Daily Papers

Transformer and its variants are fundamental neural architectures in deep learning. Recent works show that learning attention in the Fourier space can improve the long sequence learning capability of Transformers. We argue that wavelet transform shall be a better choice because it captures both position and frequency information with linear time complexity. Therefore, in this paper, we systematically study the synergy between wavelet transform and Transformers. We propose Wavelet Space Attention (WavSpA) that facilitates attention learning in a learnable wavelet coefficient space which replaces the attention in Transformers by (1) applying forward wavelet transform to project the input sequences to multi-resolution bases, (2) conducting attention learning in the wavelet coefficient space, and (3) reconstructing the representation in input space via backward wavelet transform. Extensive experiments on the Long Range Arena demonstrate that learning attention in the wavelet space using either fixed or adaptive wavelets can consistently improve Transformer's performance and also significantly outperform learning in Fourier space. We further show our method can enhance Transformer's reasoning extrapolation capability over distance on the LEGO chain-of-reasoning task.

Neural networks have shown remarkable performance in computer vision, but their deployment in numerous scientific and technical fields is challenging due to their black-box nature. Scientists and practitioners need to evaluate the reliability of a decision, i.e., to know simultaneously if a model relies on the relevant features and whether these features are robust to image corruptions. Existing attribution methods aim to provide human-understandable explanations by highlighting important regions in the image domain, but fail to fully characterize a decision process's reliability. To bridge this gap, we introduce the Wavelet sCale Attribution Method (WCAM), a generalization of attribution from the pixel domain to the space-scale domain using wavelet transforms. Attribution in the wavelet domain reveals where {\it and} on what scales the model focuses, thus enabling us to assess whether a decision is reliable.

We introduce a novel state-space architecture for diffusion models, effectively harnessing spatial and frequency information to enhance the inductive bias towards local features in input images for image generation tasks. While state-space networks, including Mamba, a revolutionary advancement in recurrent neural networks, typically scan input sequences from left to right, they face difficulties in designing effective scanning strategies, especially in the processing of image data. Our method demonstrates that integrating wavelet transformation into Mamba enhances the local structure awareness of visual inputs and better captures long-range relations of frequencies by disentangling them into wavelet subbands, representing both low- and high-frequency components. These wavelet-based outputs are then processed and seamlessly fused with the original Mamba outputs through a cross-attention fusion layer, combining both spatial and frequency information to optimize the order awareness of state-space models which is essential for the details and overall quality of image generation. Besides, we introduce a globally-shared transformer to supercharge the performance of Mamba, harnessing its exceptional power to capture global relationships. Through extensive experiments on standard benchmarks, our method demonstrates superior results compared to DiT and DIFFUSSM, achieving faster training convergence and delivering high-quality outputs. The codes and pretrained models are released at https://github.com/VinAIResearch/DiMSUM.git.

Due to the three-dimensional nature of CT- or MR-scans, generative modeling of medical images is a particularly challenging task. Existing approaches mostly apply patch-wise, slice-wise, or cascaded generation techniques to fit the high-dimensional data into the limited GPU memory. However, these approaches may introduce artifacts and potentially restrict the model's applicability for certain downstream tasks. This work presents WDM, a wavelet-based medical image synthesis framework that applies a diffusion model on wavelet decomposed images. The presented approach is a simple yet effective way of scaling diffusion models to high resolutions and can be trained on a single 40 GB GPU. Experimental results on BraTS and LIDC-IDRI unconditional image generation at a resolution of 128 times 128 times 128 show state-of-the-art image fidelity (FID) and sample diversity (MS-SSIM) scores compared to GANs, Diffusion Models, and Latent Diffusion Models. Our proposed method is the only one capable of generating high-quality images at a resolution of 256 times 256 times 256.

Implicit neural representations (INRs) have arisen as useful methods for representing signals on Euclidean domains. By parameterizing an image as a multilayer perceptron (MLP) on Euclidean space, INRs effectively represent signals in a way that couples spatial and spectral features of the signal that is not obvious in the usual discrete representation, paving the way for continuous signal processing and machine learning approaches that were not previously possible. Although INRs using sinusoidal activation functions have been studied in terms of Fourier theory, recent works have shown the advantage of using wavelets instead of sinusoids as activation functions, due to their ability to simultaneously localize in both frequency and space. In this work, we approach such INRs and demonstrate how they resolve high-frequency features of signals from coarse approximations done in the first layer of the MLP. This leads to multiple prescriptions for the design of INR architectures, including the use of complex wavelets, decoupling of low and band-pass approximations, and initialization schemes based on the singularities of the desired signal.

Source separation involves the ill-posed problem of retrieving a set of source signals that have been observed through a mixing operator. Solving this problem requires prior knowledge, which is commonly incorporated by imposing regularity conditions on the source signals, or implicitly learned through supervised or unsupervised methods from existing data. While data-driven methods have shown great promise in source separation, they often require large amounts of data, which rarely exists in planetary space missions. To address this challenge, we propose an unsupervised source separation scheme for domains with limited data access that involves solving an optimization problem in the wavelet scattering covariance representation spacex2014an interpretable, low-dimensional representation of stationary processes. We present a real-data example in which we remove transient, thermally-induced microtiltsx2014known as glitchesx2014from data recorded by a seismometer during NASA's InSight mission on Mars. Thanks to the wavelet scattering covariances' ability to capture non-Gaussian properties of stochastic processes, we are able to separate glitches using only a few glitch-free data snippets.

Image Super-Resolution is a fundamental problem in computer vision with broad applications spacing from medical imaging to satellite analysis. The ability to reconstruct high-resolution images from low-resolution inputs is crucial for enhancing downstream tasks such as object detection and segmentation. While deep learning has significantly advanced SR, achieving high-quality reconstructions with fine-grained details and realistic textures remains challenging, particularly at high upscaling factors. Recent approaches leveraging diffusion models have demonstrated promising results, yet they often struggle to balance perceptual quality with structural fidelity. In this work, we introduce ResQu a novel SR framework that integrates a quaternion wavelet preprocessing framework with latent diffusion models, incorporating a new quaternion wavelet- and time-aware encoder. Unlike prior methods that simply apply wavelet transforms within diffusion models, our approach enhances the conditioning process by exploiting quaternion wavelet embeddings, which are dynamically integrated at different stages of denoising. Furthermore, we also leverage the generative priors of foundation models such as Stable Diffusion. Extensive experiments on domain-specific datasets demonstrate that our method achieves outstanding SR results, outperforming in many cases existing approaches in perceptual quality and standard evaluation metrics. The code will be available after the revision process.

In this paper, we take a new approach to autoregressive image generation that is based on two main ingredients. The first is wavelet image coding, which allows to tokenize the visual details of an image from coarse to fine details by ordering the information starting with the most significant bits of the most significant wavelet coefficients. The second is a variant of a language transformer whose architecture is re-designed and optimized for token sequences in this 'wavelet language'. The transformer learns the significant statistical correlations within a token sequence, which are the manifestations of well-known correlations between the wavelet subbands at various resolutions. We show experimental results with conditioning on the generation process.

Transformers have emerged as a preferred model for many tasks in natural langugage processing and vision. Recent efforts on training and deploying Transformers more efficiently have identified many strategies to approximate the self-attention matrix, a key module in a Transformer architecture. Effective ideas include various prespecified sparsity patterns, low-rank basis expansions and combinations thereof. In this paper, we revisit classical Multiresolution Analysis (MRA) concepts such as Wavelets, whose potential value in this setting remains underexplored thus far. We show that simple approximations based on empirical feedback and design choices informed by modern hardware and implementation challenges, eventually yield a MRA-based approach for self-attention with an excellent performance profile across most criteria of interest. We undertake an extensive set of experiments and demonstrate that this multi-resolution scheme outperforms most efficient self-attention proposals and is favorable for both short and long sequences. Code is available at https://github.com/mlpen/mra-attention.

Time series classification (TSC) on multivariate time series is a critical problem. We propose a novel multi-view approach integrating frequency-domain and time-domain features to provide complementary contexts for TSC. Our method fuses continuous wavelet transform spectral features with temporal convolutional or multilayer perceptron features. We leverage the Mamba state space model for efficient and scalable sequence modeling. We also introduce a novel tango scanning scheme to better model sequence relationships. Experiments on 10 standard benchmark datasets demonstrate our approach achieves an average 6.45% accuracy improvement over state-of-the-art TSC models.

As ground-based all-sky astronomical surveys will gather millions of images in the coming years, a critical requirement emerges for the development of fast deconvolution algorithms capable of efficiently improving the spatial resolution of these images. By successfully recovering clean and high-resolution images from these surveys, the objective is to deepen the understanding of galaxy formation and evolution through accurate photometric measurements. We introduce a two-step deconvolution framework using a Swin Transformer architecture. Our study reveals that the deep learning-based solution introduces a bias, constraining the scope of scientific analysis. To address this limitation, we propose a novel third step relying on the active coefficients in the sparsity wavelet framework. We conducted a performance comparison between our deep learning-based method and Firedec, a classical deconvolution algorithm, based on an analysis of a subset of the EDisCS cluster samples. We demonstrate the advantage of our method in terms of resolution recovery, generalisation to different noise properties, and computational efficiency. The analysis of this cluster sample not only allowed us to assess the efficiency of our method, but it also enabled us to quantify the number of clumps within these galaxies in relation to their disc colour. This robust technique that we propose holds promise for identifying structures in the distant universe through ground-based images.

We consider the problem of approximating a function from L^2 by an element of a given m-dimensional space V_m, associated with some feature map varphi, using evaluations of the function at random points x_1,dots,x_n. After recalling some results on optimal weighted least-squares using independent and identically distributed points, we consider weighted least-squares using projection determinantal point processes (DPP) or volume sampling. These distributions introduce dependence between the points that promotes diversity in the selected features varphi(x_i). We first provide a generalized version of volume-rescaled sampling yielding quasi-optimality results in expectation with a number of samples n = O(mlog(m)), that means that the expected L^2 error is bounded by a constant times the best approximation error in L^2. Also, further assuming that the function is in some normed vector space H continuously embedded in L^2, we further prove that the approximation is almost surely bounded by the best approximation error measured in the H-norm. This includes the cases of functions from L^infty or reproducing kernel Hilbert spaces. Finally, we present an alternative strategy consisting in using independent repetitions of projection DPP (or volume sampling), yielding similar error bounds as with i.i.d. or volume sampling, but in practice with a much lower number of samples. Numerical experiments illustrate the performance of the different strategies.

Efficiently capturing the long-range patterns in sequential data sources salient to a given task -- such as classification and generative modeling -- poses a fundamental challenge. Popular approaches in the space tradeoff between the memory burden of brute-force enumeration and comparison, as in transformers, the computational burden of complicated sequential dependencies, as in recurrent neural networks, or the parameter burden of convolutional networks with many or large filters. We instead take inspiration from wavelet-based multiresolution analysis to define a new building block for sequence modeling, which we call a MultiresLayer. The key component of our model is the multiresolution convolution, capturing multiscale trends in the input sequence. Our MultiresConv can be implemented with shared filters across a dilated causal convolution tree. Thus it garners the computational advantages of convolutional networks and the principled theoretical motivation of wavelet decompositions. Our MultiresLayer is straightforward to implement, requires significantly fewer parameters, and maintains at most a O(Nlog N) memory footprint for a length N sequence. Yet, by stacking such layers, our model yields state-of-the-art performance on a number of sequence classification and autoregressive density estimation tasks using CIFAR-10, ListOps, and PTB-XL datasets.

To achieve promising results on removing noise from real-world images, most of existing denoising networks are formulated with complex network structure, making them impractical for deployment. Some attempts focused on reducing the number of filters and feature channels but suffered from large performance loss, and a more practical and lightweight denoising network with fast inference speed is of high demand. To this end, a Thumbnail based Denoising Network dubbed Thunder, is proposed and implemented as a lightweight structure for fast restoration without comprising the denoising capabilities. Specifically, the Thunder model contains two newly-established modules: (1) a wavelet-based Thumbnail Subspace Encoder (TSE) which can leverage sub-bands correlation to provide an approximate thumbnail based on the low-frequent feature; (2) a Subspace Projection based Refine Module (SPR) which can restore the details for thumbnail progressively based on the subspace projection approach. Extensive experiments have been carried out on two real-world denoising benchmarks, demonstrating that the proposed Thunder outperforms the existing lightweight models and achieves competitive performance on PSNR and SSIM when compared with the complex designs.

Low-light image enhancement techniques have significantly progressed, but unstable image quality recovery and unsatisfactory visual perception are still significant challenges. To solve these problems, we propose a novel and robust low-light image enhancement method via CLIP-Fourier Guided Wavelet Diffusion, abbreviated as CFWD. Specifically, CFWD leverages multimodal visual-language information in the frequency domain space created by multiple wavelet transforms to guide the enhancement process. Multi-scale supervision across different modalities facilitates the alignment of image features with semantic features during the wavelet diffusion process, effectively bridging the gap between degraded and normal domains. Moreover, to further promote the effective recovery of the image details, we combine the Fourier transform based on the wavelet transform and construct a Hybrid High Frequency Perception Module (HFPM) with a significant perception of the detailed features. This module avoids the diversity confusion of the wavelet diffusion process by guiding the fine-grained structure recovery of the enhancement results to achieve favourable metric and perceptually oriented enhancement. Extensive quantitative and qualitative experiments on publicly available real-world benchmarks show that our approach outperforms existing state-of-the-art methods, achieving significant progress in image quality and noise suppression. The project code is available at https://github.com/hejh8/CFWD.

Despite the recent visually-pleasing results achieved, the massive computational cost has been a long-standing flaw for diffusion probabilistic models (DPMs), which, in turn, greatly limits their applications on resource-limited platforms. Prior methods towards efficient DPM, however, have largely focused on accelerating the testing yet overlooked their huge complexity and sizes. In this paper, we make a dedicated attempt to lighten DPM while striving to preserve its favourable performance. We start by training a small-sized latent diffusion model (LDM) from scratch, but observe a significant fidelity drop in the synthetic images. Through a thorough assessment, we find that DPM is intrinsically biased against high-frequency generation, and learns to recover different frequency components at different time-steps. These properties make compact networks unable to represent frequency dynamics with accurate high-frequency estimation. Towards this end, we introduce a customized design for slim DPM, which we term as Spectral Diffusion (SD), for light-weight image synthesis. SD incorporates wavelet gating in its architecture to enable frequency dynamic feature extraction at every reverse steps, and conducts spectrum-aware distillation to promote high-frequency recovery by inverse weighting the objective based on spectrum magni tudes. Experimental results demonstrate that, SD achieves 8-18x computational complexity reduction as compared to the latent diffusion models on a series of conditional and unconditional image generation tasks while retaining competitive image fidelity.

Effective hydrological modeling and extreme weather analysis demand precipitation data at a kilometer-scale resolution, which is significantly finer than the 10 km scale offered by standard global products like IMERG. To address this, we propose the Wavelet Diffusion Model (WDM), a generative framework that achieves 10x spatial super-resolution (downscaling to 1 km) and delivers a 9x inference speedup over pixel-based diffusion models. WDM is a conditional diffusion model that learns the learns the complex structure of precipitation from MRMS radar data directly in the wavelet domain. By focusing on high-frequency wavelet coefficients, it generates exceptionally realistic and detailed 1-km precipitation fields. This wavelet-based approach produces visually superior results with fewer artifacts than pixel-space models, and delivers a significant gains in sampling efficiency. Our results demonstrate that WDM provides a robust solution to the dual challenges of accuracy and speed in geoscience super-resolution, paving the way for more reliable hydrological forecasts.

Diffusion models are rising as a powerful solution for high-fidelity image generation, which exceeds GANs in quality in many circumstances. However, their slow training and inference speed is a huge bottleneck, blocking them from being used in real-time applications. A recent DiffusionGAN method significantly decreases the models' running time by reducing the number of sampling steps from thousands to several, but their speeds still largely lag behind the GAN counterparts. This paper aims to reduce the speed gap by proposing a novel wavelet-based diffusion scheme. We extract low-and-high frequency components from both image and feature levels via wavelet decomposition and adaptively handle these components for faster processing while maintaining good generation quality. Furthermore, we propose to use a reconstruction term, which effectively boosts the model training convergence. Experimental results on CelebA-HQ, CIFAR-10, LSUN-Church, and STL-10 datasets prove our solution is a stepping-stone to offering real-time and high-fidelity diffusion models. Our code and pre-trained checkpoints are available at https://github.com/VinAIResearch/WaveDiff.git.

Food safety and quality are paramount concerns worldwide, especially concerning nutritional quality and its impact on human health. Ensuring the accuracy and efficiency of milk quality assessment is vital for maintaining the quality of dairy farm produce. Milk spectral data, Mid-infrared spectra (MIRS) of milk samples, are frequently employed for milk quality evaluations, encompassing various milk quality parameters. However, conventional milk quality analyses have overlooked the scaling nature, known as stochastic similarity in different scales, inherent in milk spectral data. Wavelet transforms are among the tools used in these analyses, although they are primarily used as data pre-processing techniques without fully realizing their potential in extracting valuable insights. The primary purpose of this study is to demonstrate the importance of accounting for scaling properties in assessing milk quality. A set of 12 descriptors is computed to characterize scaling properties in milk spectral data within the wavelet domain. These descriptors are then assessed for their effectiveness in milk quality assessments utilizing 18 different milk quality parameters. They notably demonstrated comparable performance to existing methods while utilizing fewer features when applied to an MIRS dataset. This innovative approach holds substantial promise for advancing the field of milk quality assessment, offering a means to achieve more accurate and efficient evaluations while shedding light on previously unexplored aspects of milk spectral data.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

Marine mammal communication is a complex field, hindered by the diversity of vocalizations and environmental factors. The Watkins Marine Mammal Sound Database (WMMD) is an extensive labeled dataset used in machine learning applications. However, the methods for data preparation, preprocessing, and classification found in the literature are quite disparate. This study first focuses on a brief review of the state-of-the-art benchmarks on the dataset, with an emphasis on clarifying data preparation and preprocessing methods. Subsequently, we propose the application of the Wavelet Scattering Transform (WST) in place of standard methods based on the Short-Time Fourier Transform (STFT). The study also tackles a classification task using an ad-hoc deep architecture with residual layers. We outperform the existing classification architecture by 6% in accuracy using WST and 8% using Mel spectrogram preprocessing, effectively reducing by half the number of misclassified samples, and reaching a top accuracy of 96%.

Parameter-efficient fine-tuning (PEFT) has gained widespread adoption across various applications. Among PEFT techniques, Low-Rank Adaptation (LoRA) and its extensions have emerged as particularly effective, allowing efficient model adaptation while significantly reducing computational overhead. However, existing approaches typically rely on global low-rank factorizations, which overlook local or multi-scale structure, failing to capture complex patterns in the weight updates. To address this, we propose WaRA, a novel PEFT method that leverages wavelet transforms to decompose the weight update matrix into a multi-resolution representation. By performing low-rank factorization in the wavelet domain and reconstructing updates through an inverse transform, WaRA obtains compressed adaptation parameters that harness multi-resolution analysis, enabling it to capture both coarse and fine-grained features while providing greater flexibility and sparser representations than standard LoRA. Through comprehensive experiments and analysis, we demonstrate that WaRA performs superior on diverse vision tasks, including image generation, classification, and semantic segmentation, significantly enhancing generated image quality while reducing computational complexity. Although WaRA was primarily designed for vision tasks, we further showcase its effectiveness in language tasks, highlighting its broader applicability and generalizability. The code is publicly available at GitHub{https://github.com/moeinheidari7829/WaRA}.

We present a novel type of neural fields that uses general radial bases for signal representation. State-of-the-art neural fields typically rely on grid-based representations for storing local neural features and N-dimensional linear kernels for interpolating features at continuous query points. The spatial positions of their neural features are fixed on grid nodes and cannot well adapt to target signals. Our method instead builds upon general radial bases with flexible kernel position and shape, which have higher spatial adaptivity and can more closely fit target signals. To further improve the channel-wise capacity of radial basis functions, we propose to compose them with multi-frequency sinusoid functions. This technique extends a radial basis to multiple Fourier radial bases of different frequency bands without requiring extra parameters, facilitating the representation of details. Moreover, by marrying adaptive radial bases with grid-based ones, our hybrid combination inherits both adaptivity and interpolation smoothness. We carefully designed weighting schemes to let radial bases adapt to different types of signals effectively. Our experiments on 2D image and 3D signed distance field representation demonstrate the higher accuracy and compactness of our method than prior arts. When applied to neural radiance field reconstruction, our method achieves state-of-the-art rendering quality, with small model size and comparable training speed.

Denoising Diffusion Models (DDMs) have emerged as a strong competitor to Generative Adversarial Networks (GANs). However, despite their widespread use in image synthesis and editing applications, their latent space is still not as well understood. Recently, a semantic latent space for DDMs, coined `h-space', was shown to facilitate semantic image editing in a way reminiscent of GANs. The h-space is comprised of the bottleneck activations in the DDM's denoiser across all timesteps of the diffusion process. In this paper, we explore the properties of h-space and propose several novel methods for finding meaningful semantic directions within it. We start by studying unsupervised methods for revealing interpretable semantic directions in pretrained DDMs. Specifically, we show that global latent directions emerge as the principal components in the latent space. Additionally, we provide a novel method for discovering image-specific semantic directions by spectral analysis of the Jacobian of the denoiser w.r.t. the latent code. Next, we extend the analysis by finding directions in a supervised fashion in unconditional DDMs. We demonstrate how such directions can be found by relying on either a labeled data set of real images or by annotating generated samples with a domain-specific attribute classifier. We further show how to semantically disentangle the found direction by simple linear projection. Our approaches are applicable without requiring any architectural modifications, text-based guidance, CLIP-based optimization, or model fine-tuning.

Video Variational Autoencoder (VAE) encodes videos into a low-dimensional latent space, becoming a key component of most Latent Video Diffusion Models (LVDMs) to reduce model training costs. However, as the resolution and duration of generated videos increase, the encoding cost of Video VAEs becomes a limiting bottleneck in training LVDMs. Moreover, the block-wise inference method adopted by most LVDMs can lead to discontinuities of latent space when processing long-duration videos. The key to addressing the computational bottleneck lies in decomposing videos into distinct components and efficiently encoding the critical information. Wavelet transform can decompose videos into multiple frequency-domain components and improve the efficiency significantly, we thus propose Wavelet Flow VAE (WF-VAE), an autoencoder that leverages multi-level wavelet transform to facilitate low-frequency energy flow into latent representation. Furthermore, we introduce a method called Causal Cache, which maintains the integrity of latent space during block-wise inference. Compared to state-of-the-art video VAEs, WF-VAE demonstrates superior performance in both PSNR and LPIPS metrics, achieving 2x higher throughput and 4x lower memory consumption while maintaining competitive reconstruction quality. Our code and models are available at https://github.com/PKU-YuanGroup/WF-VAE.

Watermarking the outputs of generative models is a crucial technique for tracing copyright and preventing potential harm from AI-generated content. In this paper, we introduce a novel technique called Tree-Ring Watermarking that robustly fingerprints diffusion model outputs. Unlike existing methods that perform post-hoc modifications to images after sampling, Tree-Ring Watermarking subtly influences the entire sampling process, resulting in a model fingerprint that is invisible to humans. The watermark embeds a pattern into the initial noise vector used for sampling. These patterns are structured in Fourier space so that they are invariant to convolutions, crops, dilations, flips, and rotations. After image generation, the watermark signal is detected by inverting the diffusion process to retrieve the noise vector, which is then checked for the embedded signal. We demonstrate that this technique can be easily applied to arbitrary diffusion models, including text-conditioned Stable Diffusion, as a plug-in with negligible loss in FID. Our watermark is semantically hidden in the image space and is far more robust than watermarking alternatives that are currently deployed. Code is available at github.com/YuxinWenRick/tree-ring-watermark.

Text-guided image editing using Text-to-Image (T2I) models often fails to yield satisfactory results, frequently introducing unintended modifications, such as the loss of local detail and color changes. In this paper, we analyze these failure cases and attribute them to the indiscriminate optimization across all frequency bands, even though only specific frequencies may require adjustment. To address this, we introduce a simple yet effective approach that enables the selective optimization of specific frequency bands within localized spatial regions for precise edits. Our method leverages wavelets to decompose images into different spatial resolutions across multiple frequency bands, enabling precise modifications at various levels of detail. To extend the applicability of our approach, we provide a comparative analysis of different frequency-domain techniques. Additionally, we extend our method to 3D texture editing by performing frequency decomposition on the triplane representation, enabling frequency-aware adjustments for 3D textures. Quantitative evaluations and user studies demonstrate the effectiveness of our method in producing high-quality and precise edits.

Low-rank adaptation~(LoRA) has recently gained much interest in fine-tuning foundation models. It effectively reduces the number of trainable parameters by incorporating low-rank matrices A and B to represent the weight change, i.e., Delta W=BA. Despite LoRA's progress, it faces storage challenges when handling extensive customization adaptations or larger base models. In this work, we aim to further compress trainable parameters by enjoying the powerful expressiveness of the Fourier transform. Specifically, we introduce FourierFT, which treats Delta W as a matrix in the spatial domain and learns only a small fraction of its spectral coefficients. With the trained spectral coefficients, we implement the inverse discrete Fourier transform to recover Delta W. Empirically, our FourierFT method shows comparable or better performance with fewer parameters than LoRA on various tasks, including natural language understanding, natural language generation, instruction tuning, and image classification. For example, when performing instruction tuning on the LLaMA2-7B model, FourierFT surpasses LoRA with only 0.064M trainable parameters, compared to LoRA's 33.5M. Our code is released at https://github.com/Chaos96/fourierft.

With the onset of large-scale astronomical surveys capturing millions of images, there is an increasing need to develop fast and accurate deconvolution algorithms that generalize well to different images. A powerful and accessible deconvolution method would allow for the reconstruction of a cleaner estimation of the sky. The deconvolved images would be helpful to perform photometric measurements to help make progress in the fields of galaxy formation and evolution. We propose a new deconvolution method based on the Learnlet transform. Eventually, we investigate and compare the performance of different Unet architectures and Learnlet for image deconvolution in the astrophysical domain by following a two-step approach: a Tikhonov deconvolution with a closed-form solution, followed by post-processing with a neural network. To generate our training dataset, we extract HST cutouts from the CANDELS survey in the F606W filter (V-band) and corrupt these images to simulate their blurred-noisy versions. Our numerical results based on these simulations show a detailed comparison between the considered methods for different noise levels.

Recent imitation learning policies, often framed as time series prediction tasks, directly map robotic observations-such as high-dimensional visual data and proprioception-into the action space. While time series prediction primarily relies on spatial domain modeling, the underutilization of frequency domain analysis in robotic manipulation trajectory prediction may lead to neglecting the inherent temporal information embedded within action sequences. To address this, we reframe imitation learning policies through the lens of the frequency domain and introduce the Wavelet Policy. This novel approach employs wavelet transforms (WT) for feature preprocessing and extracts multi-scale features from the frequency domain using the SE2MD (Single Encoder to Multiple Decoder) architecture. Furthermore, to enhance feature mapping in the frequency domain and increase model capacity, we introduce a Learnable Frequency-Domain Filter (LFDF) after each frequency decoder, improving adaptability under different visual conditions. Our results show that the Wavelet Policy outperforms state-of-the-art (SOTA) end-to-end methods by over 10% on four challenging robotic arm tasks, while maintaining a comparable parameter count. In long-range settings, its performance declines more slowly as task volume increases. The source code is available at https://github.com/lurenjia384/Wavelet_Policy.

U-Net and its variants have been widely used in medical image segmentation. However, most current U-Net variants confine their improvement strategies to building more complex encoder, while leaving the decoder unchanged or adopting a simple symmetric structure. These approaches overlook the true functionality of the decoder: receiving low-resolution feature maps from the encoder and restoring feature map resolution and lost information through upsampling. As a result, the decoder, especially its upsampling component, plays a crucial role in enhancing segmentation outcomes. However, in 3D medical image segmentation, the commonly used transposed convolution can result in visual artifacts. This issue stems from the absence of direct relationship between adjacent pixels in the output feature map. Furthermore, plain encoder has already possessed sufficient feature extraction capability because downsampling operation leads to the gradual expansion of the receptive field, but the loss of information during downsampling process is unignorable. To address the gap in relevant research, we extend our focus beyond the encoder and introduce neU-Net (i.e., not complex encoder U-Net), which incorporates a novel Sub-pixel Convolution for upsampling to construct a powerful decoder. Additionally, we introduce multi-scale wavelet inputs module on the encoder side to provide additional information. Our model design achieves excellent results, surpassing other state-of-the-art methods on both the Synapse and ACDC datasets.

Simulating and controlling physical systems described by partial differential equations (PDEs) are crucial tasks across science and engineering. Recently, diffusion generative models have emerged as a competitive class of methods for these tasks due to their ability to capture long-term dependencies and model high-dimensional states. However, diffusion models typically struggle with handling system states with abrupt changes and generalizing to higher resolutions. In this work, we propose Wavelet Diffusion Neural Operator (WDNO), a novel PDE simulation and control framework that enhances the handling of these complexities. WDNO comprises two key innovations. Firstly, WDNO performs diffusion-based generative modeling in the wavelet domain for the entire trajectory to handle abrupt changes and long-term dependencies effectively. Secondly, to address the issue of poor generalization across different resolutions, which is one of the fundamental tasks in modeling physical systems, we introduce multi-resolution training. We validate WDNO on five physical systems, including 1D advection equation, three challenging physical systems with abrupt changes (1D Burgers' equation, 1D compressible Navier-Stokes equation and 2D incompressible fluid), and a real-world dataset ERA5, which demonstrates superior performance on both simulation and control tasks over state-of-the-art methods, with significant improvements in long-term and detail prediction accuracy. Remarkably, in the challenging context of the 2D high-dimensional and indirect control task aimed at reducing smoke leakage, WDNO reduces the leakage by 33.2% compared to the second-best baseline. The code can be found at https://github.com/AI4Science-WestlakeU/wdno.git.

This paper explores image modeling from the frequency space and introduces DCTdiff, an end-to-end diffusion generative paradigm that efficiently models images in the discrete cosine transform (DCT) space. We investigate the design space of DCTdiff and reveal the key design factors. Experiments on different frameworks (UViT, DiT), generation tasks, and various diffusion samplers demonstrate that DCTdiff outperforms pixel-based diffusion models regarding generative quality and training efficiency. Remarkably, DCTdiff can seamlessly scale up to high-resolution generation without using the latent diffusion paradigm. Finally, we illustrate several intriguing properties of DCT image modeling. For example, we provide a theoretical proof of why `image diffusion can be seen as spectral autoregression', bridging the gap between diffusion and autoregressive models. The effectiveness of DCTdiff and the introduced properties suggest a promising direction for image modeling in the frequency space. The code is at https://github.com/forever208/DCTdiff.

Diffusion models have emerged as the leading approach for image synthesis, demonstrating exceptional photorealism and diversity. However, training diffusion models at high resolutions remains computationally prohibitive, and existing zero-shot generation techniques for synthesizing images beyond training resolutions often produce artifacts, including object duplication and spatial incoherence. In this paper, we introduce HiWave, a training-free, zero-shot approach that substantially enhances visual fidelity and structural coherence in ultra-high-resolution image synthesis using pretrained diffusion models. Our method employs a two-stage pipeline: generating a base image from the pretrained model followed by a patch-wise DDIM inversion step and a novel wavelet-based detail enhancer module. Specifically, we first utilize inversion methods to derive initial noise vectors that preserve global coherence from the base image. Subsequently, during sampling, our wavelet-domain detail enhancer retains low-frequency components from the base image to ensure structural consistency, while selectively guiding high-frequency components to enrich fine details and textures. Extensive evaluations using Stable Diffusion XL demonstrate that HiWave effectively mitigates common visual artifacts seen in prior methods, achieving superior perceptual quality. A user study confirmed HiWave's performance, where it was preferred over the state-of-the-art alternative in more than 80% of comparisons, highlighting its effectiveness for high-quality, ultra-high-resolution image synthesis without requiring retraining or architectural modifications.

Graph neural networks aim to learn representations for graph-structured data and show impressive performance, particularly in node classification. Recently, many methods have studied the representations of GNNs from the perspective of optimization goals and spectral graph theory. However, the feature space that dominates representation learning has not been systematically studied in graph neural networks. In this paper, we propose to fill this gap by analyzing the feature space of both spatial and spectral models. We decompose graph neural networks into determined feature spaces and trainable weights, providing the convenience of studying the feature space explicitly using matrix space analysis. In particular, we theoretically find that the feature space tends to be linearly correlated due to repeated aggregations. Motivated by these findings, we propose 1) feature subspaces flattening and 2) structural principal components to expand the feature space. Extensive experiments verify the effectiveness of our proposed more comprehensive feature space, with comparable inference time to the baseline, and demonstrate its efficient convergence capability.

This paper contributes to the "BraTS 2024 Brain MR Image Synthesis Challenge" and presents a conditional Wavelet Diffusion Model (cWDM) for directly solving a paired image-to-image translation task on high-resolution volumes. While deep learning-based brain tumor segmentation models have demonstrated clear clinical utility, they typically require MR scans from various modalities (T1, T1ce, T2, FLAIR) as input. However, due to time constraints or imaging artifacts, some of these modalities may be missing, hindering the application of well-performing segmentation algorithms in clinical routine. To address this issue, we propose a method that synthesizes one missing modality image conditioned on three available images, enabling the application of downstream segmentation models. We treat this paired image-to-image translation task as a conditional generation problem and solve it by combining a Wavelet Diffusion Model for high-resolution 3D image synthesis with a simple conditioning strategy. This approach allows us to directly apply our model to full-resolution volumes, avoiding artifacts caused by slice- or patch-wise data processing. While this work focuses on a specific application, the presented method can be applied to all kinds of paired image-to-image translation problems, such as CT leftrightarrow MR and MR leftrightarrow PET translation, or mask-conditioned anatomically guided image generation.

In this paper, we propose to use the general L^2-based Sobolev norms, i.e., H^s norms where sin R, to measure the data discrepancy due to noise in image processing tasks that are formulated as optimization problems. As opposed to a popular trend of developing regularization methods, we emphasize that an implicit regularization effect can be achieved through the class of Sobolev norms as the data-fitting term. Specifically, we analyze that the implicit regularization comes from the weights that the H^s norm imposes on different frequency contents of an underlying image. We further analyze the underlying noise assumption of using the Sobolev norm as the data-fitting term from a Bayesian perspective, build the connections with the Sobolev gradient-based methods and discuss the preconditioning effects on the convergence rate of the gradient descent algorithm, leading to a better understanding of functional spaces/metrics and the optimization process involved in image processing. Numerical results in full waveform inversion, image denoising and deblurring demonstrate the implicit regularization effects.

Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored the multiscale architectures and various operator designs, they are limited to learning the operators as a whole in the coordinate space. In real physical science problems, PDEs are complex coupled equations with numerical solvers relying on discretization into high-dimensional coordinate space, which cannot be precisely approximated by a single operator nor efficiently learned due to the curse of dimensionality. We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Going beyond the coordinate space, LSM enables an attention-based hierarchical projection network to reduce the high-dimensional data into a compact latent space in linear time. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space that approximates complex input-output mappings via learning multiple basis operators, enjoying nice theoretical guarantees for convergence and approximation. Experimentally, LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks covering both solid and fluid physics. Code is available at https://github.com/thuml/Latent-Spectral-Models.

Adaptive sparse coding methods learn a possibly overcomplete set of basis functions, such that natural image patches can be reconstructed by linearly combining a small subset of these bases. The applicability of these methods to visual object recognition tasks has been limited because of the prohibitive cost of the optimization algorithms required to compute the sparse representation. In this work we propose a simple and efficient algorithm to learn basis functions. After training, this model also provides a fast and smooth approximator to the optimal representation, achieving even better accuracy than exact sparse coding algorithms on visual object recognition tasks.

Although certain vision transformer (ViT) and CNN architectures generalize well on vision tasks, it is often impractical to use them on green, edge, or desktop computing due to their computational requirements for training and even testing. We present WaveMix as an alternative neural architecture that uses a multi-scale 2D discrete wavelet transform (DWT) for spatial token mixing. Unlike ViTs, WaveMix neither unrolls the image nor requires self-attention of quadratic complexity. Additionally, DWT introduces another inductive bias -- besides convolutional filtering -- to utilize the 2D structure of an image to improve generalization. The multi-scale nature of the DWT also reduces the requirement for a deeper architecture compared to the CNNs, as the latter relies on pooling for partial spatial mixing. WaveMix models show generalization that is competitive with ViTs, CNNs, and token mixers on several datasets while requiring lower GPU RAM (training and testing), number of computations, and storage. WaveMix have achieved State-of-the-art (SOTA) results in EMNIST Byclass and EMNIST Balanced datasets.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

Spectral analysis provides one of the most effective paradigms for information-preserving dimensionality reduction, as simple descriptions of naturally occurring signals are often obtained via few terms of periodic basis functions. In this work, we study deep neural networks designed to harness the structure in frequency domain for efficient learning of long-range correlations in space or time: frequency-domain models (FDMs). Existing FDMs are based on complex-valued transforms i.e. Fourier Transforms (FT), and layers that perform computation on the spectrum and input data separately. This design introduces considerable computational overhead: for each layer, a forward and inverse FT. Instead, this work introduces a blueprint for frequency domain learning through a single transform: transform once (T1). To enable efficient, direct learning in the frequency domain we derive a variance-preserving weight initialization scheme and investigate methods for frequency selection in reduced-order FDMs. Our results noticeably streamline the design process of FDMs, pruning redundant transforms, and leading to speedups of 3x to 10x that increase with data resolution and model size. We perform extensive experiments on learning the solution operator of spatio-temporal dynamics, including incompressible Navier-Stokes, turbulent flows around airfoils and high-resolution video of smoke. T1 models improve on the test performance of FDMs while requiring significantly less computation (5 hours instead of 32 for our large-scale experiment), with over 20% reduction in average predictive error across tasks.

The input space of a neural network with ReLU-like activations is partitioned into multiple linear regions, each corresponding to a specific activation pattern of the included ReLU-like activations. We demonstrate that this partition exhibits the following encoding properties across a variety of deep learning models: (1) {\it determinism}: almost every linear region contains at most one training example. We can therefore represent almost every training example by a unique activation pattern, which is parameterized by a {\it neural code}; and (2) {\it categorization}: according to the neural code, simple algorithms, such as K-Means, K-NN, and logistic regression, can achieve fairly good performance on both training and test data. These encoding properties surprisingly suggest that {\it normal neural networks well-trained for classification behave as hash encoders without any extra efforts.} In addition, the encoding properties exhibit variability in different scenarios. {Further experiments demonstrate that {\it model size}, {\it training time}, {\it training sample size}, {\it regularization}, and {\it label noise} contribute in shaping the encoding properties, while the impacts of the first three are dominant.} We then define an {\it activation hash phase chart} to represent the space expanded by {model size}, training time, training sample size, and the encoding properties, which is divided into three canonical regions: {\it under-expressive regime}, {\it critically-expressive regime}, and {\it sufficiently-expressive regime}. The source code package is available at https://github.com/LeavesLei/activation-code.

Despite the groundbreaking success of diffusion models in generating high-fidelity images, their latent space remains relatively under-explored, even though it holds significant promise for enabling versatile and interpretable image editing capabilities. The complicated denoising trajectory and high dimensionality of the latent space make it extremely challenging to interpret. Existing methods mainly explore the feature space of U-Net in Diffusion Models (DMs) instead of the latent space itself. In contrast, we directly investigate the latent space via Singular Value Decomposition (SVD) and discover three useful properties that can be used to control generation results without the requirements of data collection and maintain identity fidelity generated images. Based on these properties, we propose a novel image editing framework that is capable of learning arbitrary attributes from one pair of latent codes destined by text prompts in Stable Diffusion Models. To validate our approach, extensive experiments are conducted to demonstrate its effectiveness and flexibility in image editing. We will release our codes soon to foster further research and applications in this area.

Automatic analysis of the enormous sets of images is a critical task in life sciences. This faces many challenges such as: algorithms are highly parameterized, significant human input is intertwined, and lacking a standard meta-visualization approach. This paper proposes an alternative iterative approach for optimizing input parameters, saving time by minimizing the user involvement, and allowing for understanding the workflow of algorithms and discovering new ones. The main focus is on developing an interactive visualization technique that enables users to analyze the relationships between sampled input parameters and corresponding output. This technique is implemented as a prototype called Veni Vidi Vici, or "I came, I saw, I conquered." This strategy is inspired by the mathematical formulas of numbering computable functions and is developed atop ImageJ, a scientific image processing program. A case study is presented to investigate the proposed framework. Finally, the paper explores some potential future issues in the application of the proposed approach in parameter space analysis in visualization.

A neural architecture with randomly initialized weights, in the infinite width limit, is equivalent to a Gaussian Random Field whose covariance function is the so-called Neural Network Gaussian Process kernel (NNGP). We prove that a reproducing kernel Hilbert space (RKHS) defined by the NNGP contains only functions that can be approximated by the architecture. To achieve a certain approximation error the required number of neurons in each layer is defined by the RKHS norm of the target function. Moreover, the approximation can be constructed from a supervised dataset by a random multi-layer representation of an input vector, together with training of the last layer's weights. For a 2-layer NN and a domain equal to an n-1-dimensional sphere in {mathbb R}^n, we compare the number of neurons required by Barron's theorem and by the multi-layer features construction. We show that if eigenvalues of the integral operator of the NNGP decay slower than k^{-n-2{3}} where k is an order of an eigenvalue, then our theorem guarantees a more succinct neural network approximation than Barron's theorem. We also make some computational experiments to verify our theoretical findings. Our experiments show that realistic neural networks easily learn target functions even when both theorems do not give any guarantees.

In the domain of Biometrics, recognition systems based on iris, fingerprint or palm print scans etc. are often considered more dependable due to extremely low variance in the properties of these entities with respect to time. However, over the last decade data processing capability of computers has increased manifold, which has made real-time video content analysis possible. This shows that the need of the hour is a robust and highly automated Face Detection and Recognition algorithm with credible accuracy rate. The proposed Face Detection and Recognition system using Discrete Wavelet Transform (DWT) accepts face frames as input from a database containing images from low cost devices such as VGA cameras, webcams or even CCTV's, where image quality is inferior. Face region is then detected using properties of L*a*b* color space and only Frontal Face is extracted such that all additional background is eliminated. Further, this extracted image is converted to grayscale and its dimensions are resized to 128 x 128 pixels. DWT is then applied to entire image to obtain the coefficients. Recognition is carried out by comparison of the DWT coefficients belonging to the test image with those of the registered reference image. On comparison, Euclidean distance classifier is deployed to validate the test image from the database. Accuracy for various levels of DWT Decomposition is obtained and hence, compared.

Adapting pre-trained foundation models for various downstream tasks has been prevalent in artificial intelligence. Due to the vast number of tasks and high costs, adjusting all parameters becomes unfeasible. To mitigate this, several fine-tuning techniques have been developed to update the pre-trained model weights in a more resource-efficient manner, such as through low-rank adjustments. Yet, almost all of these methods focus on linear weights, neglecting the intricacies of parameter spaces in higher dimensions like 4D. Alternatively, some methods can be adapted for high-dimensional parameter space by compressing changes in the original space into two dimensions and then employing low-rank matrix decomposition. However, these approaches destructs the structural integrity of the involved high-dimensional spaces. To tackle the diversity of dimensional spaces across different foundation models and provide a more precise representation of the changes within these spaces, this paper introduces a generalized parameter-efficient fine-tuning framework, FLoRA, designed for various dimensional parameter space. Specifically, utilizing Tucker decomposition, FLoRA asserts that changes in each dimensional parameter space are based on a low-rank core space which maintains the consistent topological structure with the original space. It then models the changes through this core space alongside corresponding weights to reconstruct alterations in the original space. FLoRA effectively preserves the structural integrity of the change of original N-dimensional parameter space, meanwhile decomposes it via low-rank tensor decomposition. Extensive experiments on computer vision, natural language processing and multi-modal tasks validate FLoRA's effectiveness. Codes are available at https://github.com/SJTU-DeepVisionLab/FLoRA.

We explore the impact of parameter sparsity on the scaling behavior of Transformers trained on massive datasets (i.e., "foundation models"), in both vision and language domains. In this setting, we identify the first scaling law describing the relationship between weight sparsity, number of non-zero parameters, and amount of training data, which we validate empirically across model and data scales; on ViT/JFT-4B and T5/C4. These results allow us to characterize the "optimal sparsity", the sparsity level which yields the best performance for a given effective model size and training budget. For a fixed number of non-zero parameters, we identify that the optimal sparsity increases with the amount of data used for training. We also extend our study to different sparsity structures (such as the hardware-friendly n:m pattern) and strategies (such as starting from a pretrained dense model). Our findings shed light on the power and limitations of weight sparsity across various parameter and computational settings, offering both theoretical understanding and practical implications for leveraging sparsity towards computational efficiency improvements.

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new Spatiotemporal Fourier Neural Operator (SFNO) that learns maps between Bochner spaces, and a new learning framework to address these issues. This new paradigm leverages wisdom from traditional numerical PDE theory and techniques to refine the pipeline of commonly adopted end-to-end neural operator training and evaluations. Specifically, in the learning problems for the turbulent flow modeling by the Navier-Stokes Equations (NSE), the proposed architecture initiates the training with a few epochs for SFNO, concluding with the freezing of most model parameters. Then, the last linear spectral convolution layer is fine-tuned without the frequency truncation. The optimization uses a negative Sobolev norm for the first time as the loss in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is almost exact thanks to the Parseval identity. This design allows the neural operators to effectively tackle low-frequency errors while the relief of the de-aliasing filter addresses high-frequency errors. Numerical experiments on commonly used benchmarks for the 2D NSE demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers.

Currently, high-dimensional data is ubiquitous in data science, which necessitates the development of techniques to decompose and interpret such multidimensional (aka tensor) datasets. Finding a low dimensional representation of the data, that is, its inherent structure, is one of the approaches that can serve to understand the dynamics of low dimensional latent features hidden in the data. Nonnegative RESCAL is one such technique, particularly well suited to analyze self-relational data, such as dynamic networks found in international trade flows. Nonnegative RESCAL computes a low dimensional tensor representation by finding the latent space containing multiple modalities. Estimating the dimensionality of this latent space is crucial for extracting meaningful latent features. Here, to determine the dimensionality of the latent space with nonnegative RESCAL, we propose a latent dimension determination method which is based on clustering of the solutions of multiple realizations of nonnegative RESCAL decompositions. We demonstrate the performance of our model selection method on synthetic data and then we apply our method to decompose a network of international trade flows data from International Monetary Fund and validate the resulting features against empirical facts from economic literature.

The Schr\"odinger Bridge (SB) is a powerful framework for solving generative modeling tasks such as unpaired domain translation. Most SB-related research focuses on continuous data space R^{D} and leaves open theoretical and algorithmic questions about applying SB methods to discrete data, e.g, on finite spaces S^{D}. Notable examples of such sets S are codebooks of vector-quantized (VQ) representations of modern autoencoders, tokens in texts, categories of atoms in molecules, etc. In this paper, we provide a theoretical and algorithmic foundation for solving SB in discrete spaces using the recently introduced Iterative Markovian Fitting (IMF) procedure. Specifically, we theoretically justify the convergence of discrete-time IMF (D-IMF) to SB in discrete spaces. This enables us to develop a practical computational algorithm for SB which we call Categorical Schr\"odinger Bridge Matching (CSBM). We show the performance of CSBM via a series of experiments with synthetic data and VQ representations of images.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size n is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in R^K, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a ell_2 distance of at most varepsilon from the true simplex (for any varepsilon>0). Also, we theoretically show that in order to achieve this bound, it is sufficient to have ngeleft(K^2/varepsilon^2right)e^{Omegaleft(K/SNR^2right)} samples, where SNR stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as SNRgeOmegaleft(K^{1/2}right), the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in ashtiani2018nearly, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

The goal of this paper is to generate realistic audio with a lightweight and fast diffusion-based vocoder, named FreGrad. Our framework consists of the following three key components: (1) We employ discrete wavelet transform that decomposes a complicated waveform into sub-band wavelets, which helps FreGrad to operate on a simple and concise feature space, (2) We design a frequency-aware dilated convolution that elevates frequency awareness, resulting in generating speech with accurate frequency information, and (3) We introduce a bag of tricks that boosts the generation quality of the proposed model. In our experiments, FreGrad achieves 3.7 times faster training time and 2.2 times faster inference speed compared to our baseline while reducing the model size by 0.6 times (only 1.78M parameters) without sacrificing the output quality. Audio samples are available at: https://mm.kaist.ac.kr/projects/FreGrad.

Stars on the AGB can exhibit acoustic pulsation modes of different radial orders, along with non-radial modes. These pulsations are essential to the mass-loss process and influence the evolutionary pathways of AGB stars. P-L relations serve as a valuable diagnostic for understanding stellar evolution along the AGB. 3D RHD simulations provide a powerful tool for investigating pulsation phenomena driven by convective processes and their non-linear coupling with stellar oscillations. We investigate multi-mode pulsations in AGB stars using advanced 3D 'star-in-a-box' simulations with the CO5BOLD code. Signatures of these multi-mode pulsations were weak in our previous 3D models. Our focus is on identifying and characterising the various pulsation modes, examining their persistence and transitions, and comparing the results with 1D model predictions and observational data where applicable. We produced a new model grid comprising AGB stars with current masses of 0.7, 0.8, and 1,M_{odot}. Fourier analysis was applied to dynamic, time-dependent quantities to extract dominant pulsation modes and their corresponding periods. Additionally, wavelet transforms were employed to identify mode-switching behaviour over time. The models successfully reproduce the P-L sequences found in AGB stars. Mode-switching phenomena are found in both the models and wavelet analyses of observational data, allowing us to infer similarities in the underlying pulsation dynamics. These 3D simulations highlight the natural emergence of multi-mode pulsations, including both radial and non-radial modes, driven by the self-consistent interplay of convection and oscillations. Our findings underscore the value of 3D RHD models in capturing the non-linear behaviour of AGB pulsations, providing insights into mode switching, envelope structures, and potential links to episodic mass-loss events.

With generative models producing high quality images that are indistinguishable from real ones, there is growing concern regarding the malicious usage of AI-generated images. Imperceptible image watermarking is one viable solution towards such concerns. Prior watermarking methods map the image to a latent space for adding the watermark. Moreover, Latent Diffusion Models (LDM) generate the image in the latent space of a pre-trained autoencoder. We argue that this latent space can be used to integrate watermarking into the generation process. To this end, we present LaWa, an in-generation image watermarking method designed for LDMs. By using coarse-to-fine watermark embedding modules, LaWa modifies the latent space of pre-trained autoencoders and achieves high robustness against a wide range of image transformations while preserving perceptual quality of the image. We show that LaWa can also be used as a general image watermarking method. Through extensive experiments, we demonstrate that LaWa outperforms previous works in perceptual quality, robustness against attacks, and computational complexity, while having very low false positive rate. Code is available here.

The recent use of diffusion prior, enhanced by pre-trained text-image models, has markedly elevated the performance of image super-resolution (SR). To alleviate the huge computational cost required by pixel-based diffusion SR, latent-based methods utilize a feature encoder to transform the image and then implement the SR image generation in a compact latent space. Nevertheless, there are two major issues that limit the performance of latent-based diffusion. First, the compression of latent space usually causes reconstruction distortion. Second, huge computational cost constrains the parameter scale of the diffusion model. To counteract these issues, we first propose a frequency compensation module that enhances the frequency components from latent space to pixel space. The reconstruction distortion (especially for high-frequency information) can be significantly decreased. Then, we propose to use Sample-Space Mixture of Experts (SS-MoE) to achieve more powerful latent-based SR, which steadily improves the capacity of the model without a significant increase in inference costs. These carefully crafted designs contribute to performance improvements in largely explored 4x blind super-resolution benchmarks and extend to large magnification factors, i.e., 8x image SR benchmarks. The code is available at https://github.com/amandaluof/moe_sr.

As the Chinese stock market continues to evolve and its market structure grows increasingly complex, traditional quantitative trading methods are facing escalating challenges. Particularly, due to policy uncertainty and the frequent market fluctuations triggered by sudden economic events, existing models often struggle to accurately predict market dynamics. To address these challenges, this paper introduces Stockformer, a price-volume factor stock selection model that integrates wavelet transformation and a multitask self-attention network, aimed at enhancing responsiveness and predictive accuracy regarding market instabilities. Through discrete wavelet transform, Stockformer decomposes stock returns into high and low frequencies, meticulously capturing long-term market trends and short-term fluctuations, including abrupt events. Moreover, the model incorporates a Dual-Frequency Spatiotemporal Encoder and graph embedding techniques to effectively capture complex temporal and spatial relationships among stocks. Employing a multitask learning strategy, it simultaneously predicts stock returns and directional trends. Experimental results show that Stockformer outperforms existing advanced methods on multiple real stock market datasets. In strategy backtesting, Stockformer consistently demonstrates exceptional stability and reliability across market conditions-whether rising, falling, or fluctuating-particularly maintaining high performance during downturns or volatile periods, indicating a high adaptability to market fluctuations. To foster innovation and collaboration in the financial analysis sector, the Stockformer model's code has been open-sourced and is available on the GitHub repository: https://github.com/Eric991005/Multitask-Stockformer.

Convolutional networks are large linear systems divided into layers and connected by non-linear units. These units are the "articulations" that allow the network to adapt to the input. To understand how a network manages to solve a problem we must look at the articulated decisions in entirety. If we could capture the actions of non-linear units for a particular input, we would be able to replay the whole system back and forth as if it was always linear. It would also reveal the actions of non-linearities because the resulting linear system, a Linear Interpreter, depends on the input image. We introduce a hooking layer, called a LinearScope, which allows us to run the network and the linear interpreter in parallel. Its implementation is simple, flexible and efficient. From here we can make many curious inquiries: how do these linear systems look like? When the rows and columns of the transformation matrix are images, how do they look like? What type of basis do these linear transformations rely on? The answers depend on the problems presented, through which we take a tour to some popular architectures used for classification, super-resolution (SR) and image-to-image translation (I2I). For classification we observe that popular networks use a pixel-wise vote per class strategy and heavily rely on bias parameters. For SR and I2I we find that CNNs use wavelet-type basis similar to the human visual system. For I2I we reveal copy-move and template-creation strategies to generate outputs.

Super-resolution (SR) and image generation are important tasks in computer vision and are widely adopted in real-world applications. Most existing methods, however, generate images only at fixed-scale magnification and suffer from over-smoothing and artifacts. Additionally, they do not offer enough diversity of output images nor image consistency at different scales. Most relevant work applied Implicit Neural Representation (INR) to the denoising diffusion model to obtain continuous-resolution yet diverse and high-quality SR results. Since this model operates in the image space, the larger the resolution of image is produced, the more memory and inference time is required, and it also does not maintain scale-specific consistency. We propose a novel pipeline that can super-resolve an input image or generate from a random noise a novel image at arbitrary scales. The method consists of a pretrained auto-encoder, a latent diffusion model, and an implicit neural decoder, and their learning strategies. The proposed method adopts diffusion processes in a latent space, thus efficient, yet aligned with output image space decoded by MLPs at arbitrary scales. More specifically, our arbitrary-scale decoder is designed by the symmetric decoder w/o up-scaling from the pretrained auto-encoder, and Local Implicit Image Function (LIIF) in series. The latent diffusion process is learnt by the denoising and the alignment losses jointly. Errors in output images are backpropagated via the fixed decoder, improving the quality of output images. In the extensive experiments using multiple public benchmarks on the two tasks i.e. image super-resolution and novel image generation at arbitrary scales, the proposed method outperforms relevant methods in metrics of image quality, diversity and scale consistency. It is significantly better than the relevant prior-art in the inference speed and memory usage.

In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree p, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, also in this case the given matrices are ill-conditioned both in the low and high frequencies for large p. More precisely, in the fractional scenario the symbol has a single zero at 0 of order α, with α the fractional derivative order that ranges from 1 to 2, and it presents an exponential decay to zero at π for increasing p that becomes faster as α approaches 1. This translates in a mitigated conditioning in the low frequencies and in a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B-spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B-splines. Finally, we perform a numerical study of the approximation behavior of polynomial B-spline collocation. This study suggests that, in line with non-fractional diffusion problems, the approximation order for smooth solutions in the fractional case is p+2-α for even p, and p+1-α for odd p.

Identifying low-dimensional latent structures within high-dimensional data has long been a central topic in the machine learning community, driven by the need for data compression, storage, transmission, and deeper data understanding. Traditional methods, such as principal component analysis (PCA) and autoencoders (AE), operate in an unsupervised manner, ignoring label information even when it is available. In this work, we introduce a unified method capable of learning latent spaces in both unsupervised and supervised settings. We formulate the problem as a nonlinear multiple-response regression within an index model context. By applying the generalized Stein's lemma, the latent space can be estimated without knowing the nonlinear link functions. Our method can be viewed as a nonlinear generalization of PCA. Moreover, unlike AE and other neural network methods that operate as "black boxes", our approach not only offers better interpretability but also reduces computational complexity while providing strong theoretical guarantees. Comprehensive numerical experiments and real data analyses demonstrate the superior performance of our method.

Ridgelet transform has been a fundamental mathematical tool in the theoretical studies of neural networks. However, the practical applicability of ridgelet transform to conducting learning tasks was limited since its numerical implementation by conventional classical computation requires an exponential runtime exp(O(D)) as data dimension D increases. To address this problem, we develop a quantum ridgelet transform (QRT), which implements the ridgelet transform of a quantum state within a linear runtime O(D) of quantum computation. As an application, we also show that one can use QRT as a fundamental subroutine for quantum machine learning (QML) to efficiently find a sparse trainable subnetwork of large shallow wide neural networks without conducting large-scale optimization of the original network. This application discovers an efficient way in this regime to demonstrate the lottery ticket hypothesis on finding such a sparse trainable neural network. These results open an avenue of QML for accelerating learning tasks with commonly used classical neural networks.

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of R^{n} for three symmetry groups that are missing from the machine learning literature: O(n), the orthogonal group; SO(n), the special orthogonal group; and Sp(n), the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of R^{n} when the group is O(n) or SO(n), and in the symplectic basis of R^{n} when the group is Sp(n).

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

In this article, we study the consistency of the template estimation with the Fr\'echet mean in quotient spaces. The Fr\'echet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.

Subspace representation is a fundamental technique in various fields of machine learning. Analyzing a geometrical relationship among multiple subspaces is essential for understanding subspace series' temporal and/or spatial dynamics. This paper proposes the second-order difference subspace, a higher-order extension of the first-order difference subspace between two subspaces that can analyze the geometrical difference between them. As a preliminary for that, we extend the definition of the first-order difference subspace to the more general setting that two subspaces with different dimensions have an intersection. We then define the second-order difference subspace by combining the concept of first-order difference subspace and principal component subspace (Karcher mean) between two subspaces, motivated by the second-order central difference method. We can understand that the first/second-order difference subspaces correspond to the velocity and acceleration of subspace dynamics from the viewpoint of a geodesic on a Grassmann manifold. We demonstrate the validity and naturalness of our second-order difference subspace by showing numerical results on two applications: temporal shape analysis of a 3D object and time series analysis of a biometric signal.

Nowadays, with the rising number of sensors in sectors such as healthcare and industry, the problem of multivariate time series classification (MTSC) is getting increasingly relevant and is a prime target for machine and deep learning approaches. Their expanding adoption in real-world environments is causing a shift in focus from the pursuit of ever-higher prediction accuracy with complex models towards practical, deployable solutions that balance accuracy and parameters such as prediction speed. An MTSC model that has attracted attention recently is ROCKET, based on random convolutional kernels, both because of its very fast training process and its state-of-the-art accuracy. However, the large number of features it utilizes may be detrimental to inference time. Examining its theoretical background and limitations enables us to address potential drawbacks and present LightWaveS: a framework for accurate MTSC, which is fast both during training and inference. Specifically, utilizing wavelet scattering transformation and distributed feature selection, we manage to create a solution that employs just 2.5% of the ROCKET features, while achieving accuracy comparable to recent MTSC models. LightWaveS also scales well across multiple compute nodes and with the number of input channels during training. In addition, it can significantly reduce the input size and provide insight to an MTSC problem by keeping only the most useful channels. We present three versions of our algorithm and their results on distributed training time and scalability, accuracy, and inference speedup. We show that we achieve speedup ranging from 9x to 53x compared to ROCKET during inference on an edge device, on datasets with comparable accuracy.

Dimensionality reduction techniques are fundamental for analyzing and visualizing high-dimensional data. With established methods like t-SNE and PCA presenting a trade-off between representational power and interpretability. This paper introduces a novel approach that bridges this gap by combining the interpretability of linear methods with the expressiveness of non-linear transformations. The proposed algorithm constructs a non-linear mapping between high-dimensional and low-dimensional spaces through a combination of linear transformations, each weighted by Gaussian functions. This architecture enables complex non-linear transformations while preserving the interpretability advantages of linear methods, as each transformation can be analyzed independently. The resulting model provides both powerful dimensionality reduction and transparent insights into the transformed space. Techniques for interpreting the learned transformations are presented, including methods for identifying suppressed dimensions and how space is expanded and contracted. These tools enable practitioners to understand how the algorithm preserves and modifies geometric relationships during dimensionality reduction. To ensure the practical utility of this algorithm, the creation of user-friendly software packages is emphasized, facilitating its adoption in both academia and industry.

Diffusion Magnetic Resonance Imaging (dMRI) plays a crucial role in the noninvasive investigation of tissue microstructural properties and structural connectivity in the in vivo human brain. However, to effectively capture the intricate characteristics of water diffusion at various directions and scales, it is important to employ comprehensive q-space sampling. Unfortunately, this requirement leads to long scan times, limiting the clinical applicability of dMRI. To address this challenge, we propose SSOR, a Simultaneous q-Space sampling Optimization and Reconstruction framework. We jointly optimize a subset of q-space samples using a continuous representation of spherical harmonic functions and a reconstruction network. Additionally, we integrate the unique properties of diffusion magnetic resonance imaging (dMRI) in both the q-space and image domains by applying l1-norm and total-variation regularization. The experiments conducted on HCP data demonstrate that SSOR has promising strengths both quantitatively and qualitatively and exhibits robustness to noise.

Modern retrieval system often requires recomputing the representation of every piece of data in the gallery when updating to a better representation model. This process is known as backfilling and can be especially costly in the real world where the gallery often contains billions of samples. Recently, researchers have proposed the idea of Backward Compatible Training (BCT) where the new representation model can be trained with an auxiliary loss to make it backward compatible with the old representation. In this way, the new representation can be directly compared with the old representation, in principle avoiding the need for any backfilling. However, followup work shows that there is an inherent tradeoff where a backward compatible representation model cannot simultaneously maintain the performance of the new model itself. This paper reports our ``not-so-surprising'' finding that adding extra dimensions to the representation can help here. However, we also found that naively increasing the dimension of the representation did not work. To deal with this, we propose Backward-compatible Training with a novel Basis Transformation (BT^2). A basis transformation (BT) is basically a learnable set of parameters that applies an orthonormal transformation. Such a transformation possesses an important property whereby the original information contained in its input is retained in its output. We show in this paper how a BT can be utilized to add only the necessary amount of additional dimensions. We empirically verify the advantage of BT^2 over other state-of-the-art methods in a wide range of settings. We then further extend BT^2 to other challenging yet more practical settings, including significant change in model architecture (CNN to Transformers), modality change, and even a series of updates in the model architecture mimicking the evolution of deep learning models.

Initialization of parameters in deep neural networks has been shown to have a big impact on the performance of the networks (Mishkin & Matas, 2015). The initialization scheme devised by He et al, allowed convolution activations to carry a constrained mean which allowed deep networks to be trained effectively (He et al., 2015a). Orthogonal initializations and more generally orthogonal matrices in standard recurrent networks have been proved to eradicate the vanishing and exploding gradient problem (Pascanu et al., 2012). Majority of current initialization schemes do not take fully into account the intrinsic structure of the convolution operator. Using the duality of the Fourier transform and the convolution operator, Convolution Aware Initialization builds orthogonal filters in the Fourier space, and using the inverse Fourier transform represents them in the standard space. With Convolution Aware Initialization we noticed not only higher accuracy and lower loss, but faster convergence. We achieve new state of the art on the CIFAR10 dataset, and achieve close to state of the art on various other tasks.

Conventional wisdom holds that autoregressive models for image generation are typically accompanied by vector-quantized tokens. We observe that while a discrete-valued space can facilitate representing a categorical distribution, it is not a necessity for autoregressive modeling. In this work, we propose to model the per-token probability distribution using a diffusion procedure, which allows us to apply autoregressive models in a continuous-valued space. Rather than using categorical cross-entropy loss, we define a Diffusion Loss function to model the per-token probability. This approach eliminates the need for discrete-valued tokenizers. We evaluate its effectiveness across a wide range of cases, including standard autoregressive models and generalized masked autoregressive (MAR) variants. By removing vector quantization, our image generator achieves strong results while enjoying the speed advantage of sequence modeling. We hope this work will motivate the use of autoregressive generation in other continuous-valued domains and applications.

We consider the problem of computing the Walsh-Hadamard Transform (WHT) of some N-length input vector in the presence of noise, where the N-point Walsh spectrum is K-sparse with K = {O}(N^{delta}) scaling sub-linearly in the input dimension N for some 0<delta<1. Over the past decade, there has been a resurgence in research related to the computation of Discrete Fourier Transform (DFT) for some length-N input signal that has a K-sparse Fourier spectrum. In particular, through a sparse-graph code design, our earlier work on the Fast Fourier Aliasing-based Sparse Transform (FFAST) algorithm computes the K-sparse DFT in time {O}(Klog K) by taking {O}(K) noiseless samples. Inspired by the coding-theoretic design framework, Scheibler et al. proposed the Sparse Fast Hadamard Transform (SparseFHT) algorithm that elegantly computes the K-sparse WHT in the absence of noise using {O}(Klog N) samples in time {O}(Klog^2 N). However, the SparseFHT algorithm explicitly exploits the noiseless nature of the problem, and is not equipped to deal with scenarios where the observations are corrupted by noise. Therefore, a question of critical interest is whether this coding-theoretic framework can be made robust to noise. Further, if the answer is yes, what is the extra price that needs to be paid for being robust to noise? In this paper, we show, quite interestingly, that there is {\it no extra price} that needs to be paid for being robust to noise other than a constant factor. In other words, we can maintain the same sample complexity {O}(Klog N) and the computational complexity {O}(Klog^2 N) as those of the noiseless case, using our SParse Robust Iterative Graph-based Hadamard Transform (SPRIGHT) algorithm.

It is often useful to compactly summarize important properties of model parameters and training data so that they can be used later without storing and/or iterating over the entire dataset. As a specific case, we consider estimating the Function Space Distance (FSD) over a training set, i.e. the average discrepancy between the outputs of two neural networks. We propose a Linearized Activation Function TRick (LAFTR) and derive an efficient approximation to FSD for ReLU neural networks. The key idea is to approximate the architecture as a linear network with stochastic gating. Despite requiring only one parameter per unit of the network, our approach outcompetes other parametric approximations with larger memory requirements. Applied to continual learning, our parametric approximation is competitive with state-of-the-art nonparametric approximations, which require storing many training examples. Furthermore, we show its efficacy in estimating influence functions accurately and detecting mislabeled examples without expensive iterations over the entire dataset.

Enhancing the robustness of deep learning models, particularly in the realm of vision transformers (ViTs), is crucial for their real-world deployment. In this work, we provide a finetuning approach to enhance the robustness of vision transformers inspired by the concept of nullspace from linear algebra. Our investigation centers on whether a vision transformer can exhibit resilience to input variations akin to the nullspace property in linear mappings, which would imply that perturbations sampled from this nullspace do not influence the model's output when added to the input. We start from the observation that many existing ViTs satisfy this property because their patch embedding layer has a non-trivial nullspace. Then, we extend the notion of nullspace to nonlinear settings and demonstrate that it is possible to synthesize approximate nullspace elements for ViT's encoder blocks through optimization. Finally, we propose a finetuning strategy for ViTs wherein we augment the training data with synthesized approximate nullspace noise. We find that our finetuning approach significantly improves the models' robustness to both adversarial and natural image perturbations.\footnote{Code is available at: https://github.com/Liu-Hy/ns-vit.

Neural Architecture Search (NAS) has been a source of dramatic improvements in neural network design, with recent results meeting or exceeding the performance of hand-tuned architectures. However, our understanding of how to represent the search space for neural net architectures and how to search that space efficiently are both still in their infancy. We have performed an in-depth analysis to identify limitations in a widely used search space and a recent architecture search method, Differentiable Architecture Search (DARTS). These findings led us to introduce novel network blocks with a more general, balanced, and consistent design; a better-optimized Cosine Power Annealing learning rate schedule; and other improvements. Our resulting sharpDARTS search is 50% faster with a 20-30% relative improvement in final model error on CIFAR-10 when compared to DARTS. Our best single model run has 1.93% (1.98+/-0.07) validation error on CIFAR-10 and 5.5% error (5.8+/-0.3) on the recently released CIFAR-10.1 test set. To our knowledge, both are state of the art for models of similar size. This model also generalizes competitively to ImageNet at 25.1% top-1 (7.8% top-5) error. We found improvements for existing search spaces but does DARTS generalize to new domains? We propose Differentiable Hyperparameter Grid Search and the HyperCuboid search space, which are representations designed to leverage DARTS for more general parameter optimization. Here we find that DARTS fails to generalize when compared against a human's one shot choice of models. We look back to the DARTS and sharpDARTS search spaces to understand why, and an ablation study reveals an unusual generalization gap. We finally propose Max-W regularization to solve this problem, which proves significantly better than the handmade design. Code will be made available.

Kernel methods are widely used in machine learning, especially for classification problems. However, the theoretical analysis of kernel classification is still limited. This paper investigates the statistical performances of kernel classifiers. With some mild assumptions on the conditional probability eta(x)=P(Y=1mid X=x), we derive an upper bound on the classification excess risk of a kernel classifier using recent advances in the theory of kernel regression. We also obtain a minimax lower bound for Sobolev spaces, which shows the optimality of the proposed classifier. Our theoretical results can be extended to the generalization error of overparameterized neural network classifiers. To make our theoretical results more applicable in realistic settings, we also propose a simple method to estimate the interpolation smoothness of 2eta(x)-1 and apply the method to real datasets.

We propose a general framework for conditional sampling in PDE-based inverse problems, targeting the recovery of whole solutions from extremely sparse or noisy measurements. This is accomplished by a function-space diffusion model and plug-and-play guidance for conditioning. Our method first trains an unconditional discretization-agnostic denoising model using neural operator architectures. At inference, we refine the samples to satisfy sparse observation data via a gradient-based guidance mechanism. Through rigorous mathematical analysis, we extend Tweedie's formula to infinite-dimensional Hilbert spaces, providing the theoretical foundation for our posterior sampling approach. Our method (FunDPS) accurately captures posterior distributions in function spaces under minimal supervision and severe data scarcity. Across five PDE tasks with only 3% observation, our method achieves an average 32% accuracy improvement over state-of-the-art fixed-resolution diffusion baselines while reducing sampling steps by 4x. Furthermore, multi-resolution fine-tuning ensures strong cross-resolution generalizability. To the best of our knowledge, this is the first diffusion-based framework to operate independently of discretization, offering a practical and flexible solution for forward and inverse problems in the context of PDEs. Code is available at https://github.com/neuraloperator/FunDPS

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.

The seminal Fast Johnson-Lindenstrauss (Fast JL) transform by Ailon and Chazelle (SICOMP'09) embeds a set of n points in d-dimensional Euclidean space into optimal k=O(varepsilon^{-2} ln n) dimensions, while preserving all pairwise distances to within a factor (1 pm varepsilon). The Fast JL transform supports computing the embedding of a data point in O(d ln d +k ln^2 n) time, where the d ln d term comes from multiplication with a d times d Hadamard matrix and the k ln^2 n term comes from multiplication with a sparse k times d matrix. Despite the Fast JL transform being more than a decade old, it is one of the fastest dimensionality reduction techniques for many tradeoffs between varepsilon, d and n. In this work, we give a surprising new analysis of the Fast JL transform, showing that the k ln^2 n term in the embedding time can be improved to (k ln^2 n)/alpha for an alpha = Omega(min{varepsilon^{-1}ln(1/varepsilon), ln n}). The improvement follows by using an even sparser matrix. We also complement our improved analysis with a lower bound showing that our new analysis is in fact tight.

The task of manipulating real image attributes through StyleGAN inversion has been extensively researched. This process involves searching latent variables from a well-trained StyleGAN generator that can synthesize a real image, modifying these latent variables, and then synthesizing an image with the desired edits. A balance must be struck between the quality of the reconstruction and the ability to edit. Earlier studies utilized the low-dimensional W-space for latent search, which facilitated effective editing but struggled with reconstructing intricate details. More recent research has turned to the high-dimensional feature space F, which successfully inverses the input image but loses much of the detail during editing. In this paper, we introduce StyleFeatureEditor -- a novel method that enables editing in both w-latents and F-latents. This technique not only allows for the reconstruction of finer image details but also ensures their preservation during editing. We also present a new training pipeline specifically designed to train our model to accurately edit F-latents. Our method is compared with state-of-the-art encoding approaches, demonstrating that our model excels in terms of reconstruction quality and is capable of editing even challenging out-of-domain examples. Code is available at https://github.com/AIRI-Institute/StyleFeatureEditor.

Recent research on time-series self-supervised models shows great promise in learning semantic representations. However, it has been limited to small-scale datasets, e.g., thousands of temporal sequences. In this work, we make key technical contributions that are tailored to the numerical properties of time-series data and allow the model to scale to large datasets, e.g., millions of temporal sequences. We adopt the Transformer architecture by first partitioning the input into non-overlapping windows. Each window is then characterized by its normalized shape and two scalar values denoting the mean and standard deviation within each window. To embed scalar values that may possess arbitrary numerical scales to high-dimensional vectors, we propose a numerically multi-scaled embedding module enumerating all possible scales for the scalar values. The model undergoes pretraining using the proposed numerically multi-scaled embedding with a simple contrastive objective on a large-scale dataset containing over a million sequences. We study its transfer performance on a number of univariate and multivariate classification benchmarks. Our method exhibits remarkable improvement against previous representation learning approaches and establishes the new state of the art, even compared with domain-specific non-learning-based methods.

Modeling multivariate time series is a well-established problem with a wide range of applications from healthcare to financial markets. Traditional State Space Models (SSMs) are classical approaches for univariate time series modeling due to their simplicity and expressive power to represent linear dependencies. They, however, have fundamentally limited expressive power to capture non-linear dependencies, are slow in practice, and fail to model the inter-variate information flow. Despite recent attempts to improve the expressive power of SSMs by using deep structured SSMs, the existing methods are either limited to univariate time series, fail to model complex patterns (e.g., seasonal patterns), fail to dynamically model the dependencies of variate and time dimensions, and/or are input-independent. We present Chimera that uses two input-dependent 2-D SSM heads with different discretization processes to learn long-term progression and seasonal patterns. To improve the efficiency of complex 2D recurrence, we present a fast training using a new 2-dimensional parallel selective scan. We further present and discuss 2-dimensional Mamba and Mamba-2 as the spacial cases of our 2D SSM. Our experimental evaluation shows the superior performance of Chimera on extensive and diverse benchmarks, including ECG and speech time series classification, long-term and short-term time series forecasting, and time series anomaly detection.

Measuring the acoustic characteristics of a space is often done by capturing its impulse response (IR), a representation of how a full-range stimulus sound excites it. This work generates an IR from a single image, which can then be applied to other signals using convolution, simulating the reverberant characteristics of the space shown in the image. Recording these IRs is both time-intensive and expensive, and often infeasible for inaccessible locations. We use an end-to-end neural network architecture to generate plausible audio impulse responses from single images of acoustic environments. We evaluate our method both by comparisons to ground truth data and by human expert evaluation. We demonstrate our approach by generating plausible impulse responses from diverse settings and formats including well known places, musical halls, rooms in paintings, images from animations and computer games, synthetic environments generated from text, panoramic images, and video conference backgrounds.

A challenging problem in many modern machine learning tasks is to process weight-space features, i.e., to transform or extract information from the weights and gradients of a neural network. Recent works have developed promising weight-space models that are equivariant to the permutation symmetries of simple feedforward networks. However, they are not applicable to general architectures, since the permutation symmetries of a weight space can be complicated by recurrence or residual connections. This work proposes an algorithm that automatically constructs permutation equivariant models, which we refer to as universal neural functionals (UNFs), for any weight space. Among other applications, we demonstrate how UNFs can be substituted into existing learned optimizer designs, and find promising improvements over prior methods when optimizing small image classifiers and language models. Our results suggest that learned optimizers can benefit from considering the (symmetry) structure of the weight space they optimize. We open-source our library for constructing UNFs at https://github.com/AllanYangZhou/universal_neural_functional.

We present CartoonX (Cartoon Explanation), a novel model-agnostic explanation method tailored towards image classifiers and based on the rate-distortion explanation (RDE) framework. Natural images are roughly piece-wise smooth signals -- also called cartoon-like images -- and tend to be sparse in the wavelet domain. CartoonX is the first explanation method to exploit this by requiring its explanations to be sparse in the wavelet domain, thus extracting the relevant piece-wise smooth part of an image instead of relevant pixel-sparse regions. We demonstrate that CartoonX can reveal novel valuable explanatory information, particularly for misclassifications. Moreover, we show that CartoonX achieves a lower distortion with fewer coefficients than other state-of-the-art methods.

Convolutional Neural Networks (CNN) have been successful in processing data signals that are uniformly sampled in the spatial domain (e.g., images). However, most data signals do not natively exist on a grid, and in the process of being sampled onto a uniform physical grid suffer significant aliasing error and information loss. Moreover, signals can exist in different topological structures as, for example, points, lines, surfaces and volumes. It has been challenging to analyze signals with mixed topologies (for example, point cloud with surface mesh). To this end, we develop mathematical formulations for Non-Uniform Fourier Transforms (NUFT) to directly, and optimally, sample nonuniform data signals of different topologies defined on a simplex mesh into the spectral domain with no spatial sampling error. The spectral transform is performed in the Euclidean space, which removes the translation ambiguity from works on the graph spectrum. Our representation has four distinct advantages: (1) the process causes no spatial sampling error during the initial sampling, (2) the generality of this approach provides a unified framework for using CNNs to analyze signals of mixed topologies, (3) it allows us to leverage state-of-the-art backbone CNN architectures for effective learning without having to design a particular architecture for a particular data structure in an ad-hoc fashion, and (4) the representation allows weighted meshes where each element has a different weight (i.e., texture) indicating local properties. We achieve results on par with the state-of-the-art for the 3D shape retrieval task, and a new state-of-the-art for the point cloud to surface reconstruction task.

Learning in weight spaces, where neural networks process the weights of other deep neural networks, has emerged as a promising research direction with applications in various fields, from analyzing and editing neural fields and implicit neural representations, to network pruning and quantization. Recent works designed architectures for effective learning in that space, which takes into account its unique, permutation-equivariant, structure. Unfortunately, so far these architectures suffer from severe overfitting and were shown to benefit from large datasets. This poses a significant challenge because generating data for this learning setup is laborious and time-consuming since each data sample is a full set of network weights that has to be trained. In this paper, we address this difficulty by investigating data augmentations for weight spaces, a set of techniques that enable generating new data examples on the fly without having to train additional input weight space elements. We first review several recently proposed data augmentation schemes %that were proposed recently and divide them into categories. We then introduce a novel augmentation scheme based on the Mixup method. We evaluate the performance of these techniques on existing benchmarks as well as new benchmarks we generate, which can be valuable for future studies.

We describe a novel attribution method which is grounded in Sensitivity Analysis and uses Sobol indices. Beyond modeling the individual contributions of image regions, Sobol indices provide an efficient way to capture higher-order interactions between image regions and their contributions to a neural network's prediction through the lens of variance. We describe an approach that makes the computation of these indices efficient for high-dimensional problems by using perturbation masks coupled with efficient estimators to handle the high dimensionality of images. Importantly, we show that the proposed method leads to favorable scores on standard benchmarks for vision (and language models) while drastically reducing the computing time compared to other black-box methods -- even surpassing the accuracy of state-of-the-art white-box methods which require access to internal representations. Our code is freely available: https://github.com/fel-thomas/Sobol-Attribution-Method

Recently, several models based on deep neural networks have achieved great success in terms of both reconstruction accuracy and computational performance for single image super-resolution. In these methods, the low resolution (LR) input image is upscaled to the high resolution (HR) space using a single filter, commonly bicubic interpolation, before reconstruction. This means that the super-resolution (SR) operation is performed in HR space. We demonstrate that this is sub-optimal and adds computational complexity. In this paper, we present the first convolutional neural network (CNN) capable of real-time SR of 1080p videos on a single K2 GPU. To achieve this, we propose a novel CNN architecture where the feature maps are extracted in the LR space. In addition, we introduce an efficient sub-pixel convolution layer which learns an array of upscaling filters to upscale the final LR feature maps into the HR output. By doing so, we effectively replace the handcrafted bicubic filter in the SR pipeline with more complex upscaling filters specifically trained for each feature map, whilst also reducing the computational complexity of the overall SR operation. We evaluate the proposed approach using images and videos from publicly available datasets and show that it performs significantly better (+0.15dB on Images and +0.39dB on Videos) and is an order of magnitude faster than previous CNN-based methods.

The k-means++ seeding algorithm (Arthur & Vassilvitskii, 2007) is widely used in practice for the k-means clustering problem where the goal is to cluster a dataset X subset R ^d into k clusters. The popularity of this algorithm is due to its simplicity and provable guarantee of being O(log k) competitive with the optimal solution in expectation. However, its running time is O(|X|kd), making it expensive for large datasets. In this work, we present a simple and effective rejection sampling based approach for speeding up k-means++. Our first method runs in time O(nnz (X) + beta k^2d) while still being O(log k ) competitive in expectation. Here, beta is a parameter which is the ratio of the variance of the dataset to the optimal k-means cost in expectation and O hides logarithmic factors in k and |X|. Our second method presents a new trade-off between computational cost and solution quality. It incurs an additional scale-invariant factor of k^{-Omega( m/beta)} Var (X) in addition to the O(log k) guarantee of k-means++ improving upon a result of (Bachem et al, 2016a) who get an additional factor of m^{-1}Var(X) while still running in time O(nnz(X) + mk^2d). We perform extensive empirical evaluations to validate our theoretical results and to show the effectiveness of our approach on real datasets.

Given alphain(0,1] and pin[1,+infty], we define the space DM^{alpha,p}(mathbb R^n) of L^p vector fields whose alpha-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the alpha-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss-Green formula. The sharpness of our results is discussed via some explicit examples.

Image-to-Image translation (I2I) is a subtype of Machine Learning (ML) that has tremendous potential in applications where two domains of images and the need for translation between the two exist, such as the removal of fog. For example, this could be useful for autonomous vehicles, which currently struggle with adverse weather conditions like fog. However, datasets for I2I tasks are not abundant and typically hard to acquire. Here, we introduce STEREOFOG, a dataset comprised of 10,067 paired fogged and clear images, captured using a custom-built device, with the purpose of exploring I2I's potential in this domain. It is the only real-world dataset of this kind to the best of our knowledge. Furthermore, we apply and optimize the pix2pix I2I ML framework to this dataset. With the final model achieving an average Complex Wavelet-Structural Similarity (CW-SSIM) score of 0.76, we prove the technique's suitability for the problem.

Finding the mode of a high dimensional probability distribution D is a fundamental algorithmic problem in statistics and data analysis. There has been particular interest in efficient methods for solving the problem when D is represented as a mixture model or kernel density estimate, although few algorithmic results with worst-case approximation and runtime guarantees are known. In this work, we significantly generalize a result of (LeeLiMusco:2021) on mode approximation for Gaussian mixture models. We develop randomized dimensionality reduction methods for mixtures involving a broader class of kernels, including the popular logistic, sigmoid, and generalized Gaussian kernels. As in Lee et al.'s work, our dimensionality reduction results yield quasi-polynomial algorithms for mode finding with multiplicative accuracy (1-epsilon) for any epsilon > 0. Moreover, when combined with gradient descent, they yield efficient practical heuristics for the problem. In addition to our positive results, we prove a hardness result for box kernels, showing that there is no polynomial time algorithm for finding the mode of a kernel density estimate, unless P = NP. Obtaining similar hardness results for kernels used in practice (like Gaussian or logistic kernels) is an interesting future direction.

Unsupervised learning of probabilistic models is a central yet challenging problem in machine learning. Specifically, designing models with tractable learning, sampling, inference and evaluation is crucial in solving this task. We extend the space of such models using real-valued non-volume preserving (real NVP) transformations, a set of powerful invertible and learnable transformations, resulting in an unsupervised learning algorithm with exact log-likelihood computation, exact sampling, exact inference of latent variables, and an interpretable latent space. We demonstrate its ability to model natural images on four datasets through sampling, log-likelihood evaluation and latent variable manipulations.

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.

We present a novel image inversion framework and a training pipeline to achieve high-fidelity image inversion with high-quality attribute editing. Inverting real images into StyleGAN's latent space is an extensively studied problem, yet the trade-off between the image reconstruction fidelity and image editing quality remains an open challenge. The low-rate latent spaces are limited in their expressiveness power for high-fidelity reconstruction. On the other hand, high-rate latent spaces result in degradation in editing quality. In this work, to achieve high-fidelity inversion, we learn residual features in higher latent codes that lower latent codes were not able to encode. This enables preserving image details in reconstruction. To achieve high-quality editing, we learn how to transform the residual features for adapting to manipulations in latent codes. We train the framework to extract residual features and transform them via a novel architecture pipeline and cycle consistency losses. We run extensive experiments and compare our method with state-of-the-art inversion methods. Qualitative metrics and visual comparisons show significant improvements. Code: https://github.com/hamzapehlivan/StyleRes

Compressed Sensing MRI reconstructs images of the body's internal anatomy from undersampled measurements, thereby reducing scan time. Recently, deep learning has shown great potential for reconstructing high-fidelity images from highly undersampled measurements. However, one needs to train multiple models for different undersampling patterns and desired output image resolutions, since most networks operate on a fixed discretization. Such approaches are highly impractical in clinical settings, where undersampling patterns and image resolutions are frequently changed to accommodate different real-time imaging and diagnostic requirements. We propose a unified MRI reconstruction model robust to various measurement undersampling patterns and image resolutions. Our approach uses neural operators, a discretization-agnostic architecture applied in both image and measurement spaces, to capture local and global features. Empirically, our model improves SSIM by 11% and PSNR by 4 dB over a state-of-the-art CNN (End-to-End VarNet), with 600times faster inference than diffusion methods. The resolution-agnostic design also enables zero-shot super-resolution and extended field-of-view reconstruction, offering a versatile and efficient solution for clinical MR imaging. Our unified model offers a versatile solution for MRI, adapting seamlessly to various measurement undersampling and imaging resolutions, making it highly effective for flexible and reliable clinical imaging. Our code is available at https://armeet.ca/nomri.

We investigate the expressive power of depth-2 bandlimited random neural networks. A random net is a neural network where the hidden layer parameters are frozen with random assignment, and only the output layer parameters are trained by loss minimization. Using random weights for a hidden layer is an effective method to avoid non-convex optimization in standard gradient descent learning. It has also been adopted in recent deep learning theories. Despite the well-known fact that a neural network is a universal approximator, in this study, we mathematically show that when hidden parameters are distributed in a bounded domain, the network may not achieve zero approximation error. In particular, we derive a new nontrivial approximation error lower bound. The proof utilizes the technique of ridgelet analysis, a harmonic analysis method designed for neural networks. This method is inspired by fundamental principles in classical signal processing, specifically the idea that signals with limited bandwidth may not always be able to perfectly recreate the original signal. We corroborate our theoretical results with various simulation studies, and generally, two main take-home messages are offered: (i) Not any distribution for selecting random weights is feasible to build a universal approximator; (ii) A suitable assignment of random weights exists but to some degree is associated with the complexity of the target function.

Popular parameter-efficient fine-tuning (PEFT) methods, such as LoRA and its variants, freeze pre-trained model weights \(W\) and inject learnable matrices \(\Delta W\). These \(\Delta W\) matrices are structured for efficient parameterization, often using techniques like low-rank approximations or scaling vectors. However, these methods typically show a performance gap compared to full fine-tuning. Although recent PEFT methods have narrowed this gap, they do so at the cost of additional learnable parameters. We propose SVFT, a simple approach that fundamentally differs from existing methods: the structure imposed on \(\Delta W\) depends on the specific weight matrix \(W\). Specifically, SVFT updates \(W\) as a sparse combination of outer products of its singular vectors, training only the coefficients (scales) of these sparse combinations. This approach allows fine-grained control over expressivity through the number of coefficients. Extensive experiments on language and vision benchmarks show that SVFT recovers up to 96% of full fine-tuning performance while training only 0.006 to 0.25% of parameters, outperforming existing methods that only recover up to 85% performance using 0.03 to 0.8% of the trainable parameter budget.

We present ConDiff, a novel dataset for scientific machine learning. ConDiff focuses on the parametric diffusion equation with space dependent coefficients, a fundamental problem in many applications of partial differential equations (PDEs). The main novelty of the proposed dataset is that we consider discontinuous coefficients with high contrast. These coefficient functions are sampled from a selected set of distributions. This class of problems is not only of great academic interest, but is also the basis for describing various environmental and industrial problems. In this way, ConDiff shortens the gap with real-world problems while remaining fully synthetic and easy to use. ConDiff consists of a diverse set of diffusion equations with coefficients covering a wide range of contrast levels and heterogeneity with a measurable complexity metric for clearer comparison between different coefficient functions. We baseline ConDiff on standard deep learning models in the field of scientific machine learning. By providing a large number of problem instances, each with its own coefficient function and right-hand side, we hope to encourage the development of novel physics-based deep learning approaches, such as neural operators, ultimately driving progress towards more accurate and efficient solutions of complex PDE problems.

This paper considers the problem of undersampled MRI reconstruction. We propose a novel Transformer-based framework for directly processing signal in k-space, going beyond the limitation of regular grids as ConvNets do. We adopt an implicit representation of k-space spectrogram, treating spatial coordinates as inputs, and dynamically query the sparsely sampled points to reconstruct the spectrogram, i.e. learning the inductive bias in k-space. To strike a balance between computational cost and reconstruction quality, we build the decoder with hierarchical structure to generate low-resolution and high-resolution outputs respectively. To validate the effectiveness of our proposed method, we have conducted extensive experiments on two public datasets, and demonstrate superior or comparable performance to state-of-the-art approaches.

We propose the use of low bit-depth Sigma-Delta and distributed noise-shaping methods for quantizing the Random Fourier features (RFFs) associated with shift-invariant kernels. We prove that our quantized RFFs -- even in the case of 1-bit quantization -- allow a high accuracy approximation of the underlying kernels, and the approximation error decays at least polynomially fast as the dimension of the RFFs increases. We also show that the quantized RFFs can be further compressed, yielding an excellent trade-off between memory use and accuracy. Namely, the approximation error now decays exponentially as a function of the bits used. Moreover, we empirically show by testing the performance of our methods on several machine learning tasks that our method compares favorably to other state of the art quantization methods in this context.

Learning physical simulations has been an essential and central aspect of many recent research efforts in machine learning, particularly for Navier-Stokes-based fluid mechanics. Classic numerical solvers have traditionally been computationally expensive and challenging to use in inverse problems, whereas Neural solvers aim to address both concerns through machine learning. We propose a general formulation for continuous convolutions using separable basis functions as a superset of existing methods and evaluate a large set of basis functions in the context of (a) a compressible 1D SPH simulation, (b) a weakly compressible 2D SPH simulation, and (c) an incompressible 2D SPH Simulation. We demonstrate that even and odd symmetries included in the basis functions are key aspects of stability and accuracy. Our broad evaluation shows that Fourier-based continuous convolutions outperform all other architectures regarding accuracy and generalization. Finally, using these Fourier-based networks, we show that prior inductive biases, such as window functions, are no longer necessary. An implementation of our approach, as well as complete datasets and solver implementations, is available at https://github.com/tum-pbs/SFBC.

Since the introduction of deep learning, a wide scope of representation properties, such as decorrelation, whitening, disentanglement, rank, isotropy, and mutual information, have been studied to improve the quality of representation. However, manipulating such properties can be challenging in terms of implementational effectiveness and general applicability. To address these limitations, we propose to regularize von Neumann entropy~(VNE) of representation. First, we demonstrate that the mathematical formulation of VNE is superior in effectively manipulating the eigenvalues of the representation autocorrelation matrix. Then, we demonstrate that it is widely applicable in improving state-of-the-art algorithms or popular benchmark algorithms by investigating domain-generalization, meta-learning, self-supervised learning, and generative models. In addition, we formally establish theoretical connections with rank, disentanglement, and isotropy of representation. Finally, we provide discussions on the dimension control of VNE and the relationship with Shannon entropy. Code is available at: https://github.com/jaeill/CVPR23-VNE.

Physics-Informed Neural Networks (PINNs) have emerged as a promising deep learning framework for approximating numerical solutions to partial differential equations (PDEs). However, conventional PINNs, relying on multilayer perceptrons (MLP), neglect the crucial temporal dependencies inherent in practical physics systems and thus fail to propagate the initial condition constraints globally and accurately capture the true solutions under various scenarios. In this paper, we introduce a novel Transformer-based framework, termed PINNsFormer, designed to address this limitation. PINNsFormer can accurately approximate PDE solutions by utilizing multi-head attention mechanisms to capture temporal dependencies. PINNsFormer transforms point-wise inputs into pseudo sequences and replaces point-wise PINNs loss with a sequential loss. Additionally, it incorporates a novel activation function, Wavelet, which anticipates Fourier decomposition through deep neural networks. Empirical results demonstrate that PINNsFormer achieves superior generalization ability and accuracy across various scenarios, including PINNs failure modes and high-dimensional PDEs. Moreover, PINNsFormer offers flexibility in integrating existing learning schemes for PINNs, further enhancing its performance.

A fuzzy theoretic analytical approach was recently introduced that leads to efficient and robust models while addressing automatically the typical issues associated to parametric deep models. However, a formal conceptualization of the fuzzy theoretic analytical deep models is still not available. This paper introduces using measure theoretic basis the notion of membership-mapping for representing data points through attribute values (motivated by fuzzy theory). A property of the membership-mapping, that can be exploited for data representation learning, is of providing an interpolation on the given data points in the data space. An analytical approach to the variational learning of a membership-mappings based data representation model is considered.

Deep learning has been wildly successful in practice and most state-of-the-art machine learning methods are based on neural networks. Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep neural networks. In this article, we present a relatively new mathematical framework that provides the beginning of a deeper understanding of deep learning. This framework precisely characterizes the functional properties of neural networks that are trained to fit to data. The key mathematical tools which support this framework include transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory, which are all techniques deeply rooted in signal processing. This framework explains the effect of weight decay regularization in neural network training, the use of skip connections and low-rank weight matrices in network architectures, the role of sparsity in neural networks, and explains why neural networks can perform well in high-dimensional problems.

Time series modeling is a well-established problem, which often requires that methods (1) expressively represent complicated dependencies, (2) forecast long horizons, and (3) efficiently train over long sequences. State-space models (SSMs) are classical models for time series, and prior works combine SSMs with deep learning layers for efficient sequence modeling. However, we find fundamental limitations with these prior approaches, proving their SSM representations cannot express autoregressive time series processes. We thus introduce SpaceTime, a new state-space time series architecture that improves all three criteria. For expressivity, we propose a new SSM parameterization based on the companion matrix -- a canonical representation for discrete-time processes -- which enables SpaceTime's SSM layers to learn desirable autoregressive processes. For long horizon forecasting, we introduce a "closed-loop" variation of the companion SSM, which enables SpaceTime to predict many future time-steps by generating its own layer-wise inputs. For efficient training and inference, we introduce an algorithm that reduces the memory and compute of a forward pass with the companion matrix. With sequence length ell and state-space size d, we go from O(d ell) na\"ively to O(d + ell). In experiments, our contributions lead to state-of-the-art results on extensive and diverse benchmarks, with best or second-best AUROC on 6 / 7 ECG and speech time series classification, and best MSE on 14 / 16 Informer forecasting tasks. Furthermore, we find SpaceTime (1) fits AR(p) processes that prior deep SSMs fail on, (2) forecasts notably more accurately on longer horizons than prior state-of-the-art, and (3) speeds up training on real-world ETTh1 data by 73% and 80% relative wall-clock time over Transformers and LSTMs.

Generating time series data using Generative Adversarial Networks (GANs) presents several prevalent challenges, such as slow convergence, information loss in embedding spaces, instability, and performance variability depending on the series length. To tackle these obstacles, we introduce a robust framework aimed at addressing and mitigating these issues effectively. This advanced framework integrates the benefits of an Autoencoder-generated embedding space with the adversarial training dynamics of GANs. This framework benefits from a time series-based loss function and oversight from a supervisory network, both of which capture the stepwise conditional distributions of the data effectively. The generator functions within the latent space, while the discriminator offers essential feedback based on the feature space. Moreover, we introduce an early generation algorithm and an improved neural network architecture to enhance stability and ensure effective generalization across both short and long time series. Through joint training, our framework consistently outperforms existing benchmarks, generating high-quality time series data across a range of real and synthetic datasets with diverse characteristics.

We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral coefficients. In contrast to current machine learning approaches which enforce PDE constraints by minimizing the numerical quadrature of the residuals in the spatiotemporal domain, we leverage Parseval's identity and introduce a new training strategy through a spectral loss. Our spectral loss enables more efficient differentiation through the neural network, and substantially reduces training complexity. At inference time, the computational cost of our method remains constant, regardless of the spatiotemporal resolution of the domain. Our experimental results demonstrate that our method significantly outperforms previous machine learning approaches in terms of speed and accuracy by one to two orders of magnitude on multiple different problems. When compared to numerical solvers of the same accuracy, our method demonstrates a 10times increase in performance speed.

Deep learning models like Transformers and Convolutional Neural Networks (CNNs) have revolutionized various domains, but their parameter-intensive nature hampers deployment in resource-constrained settings. In this paper, we introduce a novel concept utilizes column space and row space of weight matrices, which allows for a substantial reduction in model parameters without compromising performance. Leveraging this paradigm, we achieve parameter-efficient deep learning models.. Our approach applies to both Bottleneck and Attention layers, effectively halving the parameters while incurring only minor performance degradation. Extensive experiments conducted on the ImageNet dataset with ViT and ResNet50 demonstrate the effectiveness of our method, showcasing competitive performance when compared to traditional models. This approach not only addresses the pressing demand for parameter efficient deep learning solutions but also holds great promise for practical deployment in real-world scenarios.

In the misspecified kernel ridge regression problem, researchers usually assume the underground true function f_{rho}^{*} in [H]^{s}, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) H for some sin (0,1). The existing minimax optimal results require |f_{rho}^{*}|_{L^{infty}}<infty which implicitly requires s > alpha_{0} where alpha_{0}in (0,1) is the embedding index, a constant depending on H. Whether the KRR is optimal for all sin (0,1) is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any sin (0,1) when the H is a Sobolev RKHS.

Recently, Visual Autoregressive (VAR) Models introduced a groundbreaking advancement in the field of image generation, offering a scalable approach through a coarse-to-fine "next-scale prediction" paradigm. However, the state-of-the-art algorithm of VAR models in [Tian, Jiang, Yuan, Peng and Wang, NeurIPS 2024] takes O(n^4) time, which is computationally inefficient. In this work, we analyze the computational limits and efficiency criteria of VAR Models through a fine-grained complexity lens. Our key contribution is identifying the conditions under which VAR computations can achieve sub-quadratic time complexity. Specifically, we establish a critical threshold for the norm of input matrices used in VAR attention mechanisms. Above this threshold, assuming the Strong Exponential Time Hypothesis (SETH) from fine-grained complexity theory, a sub-quartic time algorithm for VAR models is impossible. To substantiate our theoretical findings, we present efficient constructions leveraging low-rank approximations that align with the derived criteria. This work initiates the study of the computational efficiency of the VAR model from a theoretical perspective. Our technique will shed light on advancing scalable and efficient image generation in VAR frameworks.

Image restoration models are typically trained with a pixel-wise distance loss defined over the RGB color representation space, which is well known to be a source of blurry and unrealistic textures in the restored images. The reason, we believe, is that the three-channel RGB space is insufficient for supervising the restoration models. To this end, we augment the representation to hold structural information of local neighborhoods at each pixel while keeping the color information and pixel-grainedness unharmed. The result is a new representation space, dubbed augmented RGB (aRGB) space. Substituting the underlying representation space for the per-pixel losses facilitates the training of image restoration models, thereby improving the performance without affecting the evaluation phase. Notably, when combined with auxiliary objectives such as adversarial or perceptual losses, our aRGB space consistently improves overall metrics by reconstructing both color and local structures, overcoming the conventional perception-distortion trade-off.

Advances in latent diffusion models (LDMs) have revolutionized high-resolution image generation, but the design space of the autoencoder that is central to these systems remains underexplored. In this paper, we introduce LiteVAE, a family of autoencoders for LDMs that leverage the 2D discrete wavelet transform to enhance scalability and computational efficiency over standard variational autoencoders (VAEs) with no sacrifice in output quality. We also investigate the training methodologies and the decoder architecture of LiteVAE and propose several enhancements that improve the training dynamics and reconstruction quality. Our base LiteVAE model matches the quality of the established VAEs in current LDMs with a six-fold reduction in encoder parameters, leading to faster training and lower GPU memory requirements, while our larger model outperforms VAEs of comparable complexity across all evaluated metrics (rFID, LPIPS, PSNR, and SSIM).

It is well noted that coordinate based MLPs benefit -- in terms of preserving high-frequency information -- through the encoding of coordinate positions as an array of Fourier features. Hitherto, the rationale for the effectiveness of these positional encodings has been solely studied through a Fourier lens. In this paper, we strive to broaden this understanding by showing that alternative non-Fourier embedding functions can indeed be used for positional encoding. Moreover, we show that their performance is entirely determined by a trade-off between the stable rank of the embedded matrix and the distance preservation between embedded coordinates. We further establish that the now ubiquitous Fourier feature mapping of position is a special case that fulfills these conditions. Consequently, we present a more general theory to analyze positional encoding in terms of shifted basis functions. To this end, we develop the necessary theoretical formulae and empirically verify that our theoretical claims hold in practice. Codes available at https://github.com/osiriszjq/Rethinking-positional-encoding.

Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.

This paper studies sequence modeling for prediction tasks with long range dependencies. We propose a new formulation for state space models (SSMs) based on learning linear dynamical systems with the spectral filtering algorithm (Hazan et al. (2017)). This gives rise to a novel sequence prediction architecture we call a spectral state space model. Spectral state space models have two primary advantages. First, they have provable robustness properties as their performance depends on neither the spectrum of the underlying dynamics nor the dimensionality of the problem. Second, these models are constructed with fixed convolutional filters that do not require learning while still outperforming SSMs in both theory and practice. The resulting models are evaluated on synthetic dynamical systems and long-range prediction tasks of various modalities. These evaluations support the theoretical benefits of spectral filtering for tasks requiring very long range memory.

In recent years, there has been a growing interest in deep learning-based pansharpening. Thus far, research has mainly focused on architectures. Nonetheless, model training is an equally important issue. A first problem is the absence of ground truths, unavoidable in pansharpening. This is often addressed by training networks in a reduced resolution domain and using the original data as ground truth, relying on an implicit scale invariance assumption. However, on full resolution images results are often disappointing, suggesting such invariance not to hold. A further problem is the scarcity of training data, which causes a limited generalization ability and a poor performance on off-training test images. In this paper, we propose a full-resolution training framework for deep learning-based pansharpening. The framework is fully general and can be used for any deep learning-based pansharpening model. Training takes place in the high-resolution domain, relying only on the original data, thus avoiding any loss of information. To ensure spectral and spatial fidelity, a suitable two-component loss is defined. The spectral component enforces consistency between the pansharpened output and the low-resolution multispectral input. The spatial component, computed at high-resolution, maximizes the local correlation between each pansharpened band and the panchromatic input. At testing time, the target-adaptive operating modality is adopted, achieving good generalization with a limited computational overhead. Experiments carried out on WorldView-3, WorldView-2, and GeoEye-1 images show that methods trained with the proposed framework guarantee a pretty good performance in terms of both full-resolution numerical indexes and visual quality.

From the Bayesian perspective, the category of conditional probabilities (a variant of the Kleisli category of the Giry monad, whose objects are measurable spaces and arrows are Markov kernels) gives a nice framework for conceptualization and analysis of many aspects of machine learning. Using categorical methods, we construct models for parametric and nonparametric Bayesian reasoning on function spaces, thus providing a basis for the supervised learning problem. In particular, stochastic processes are arrows to these function spaces which serve as prior probabilities. The resulting inference maps can often be analytically constructed in this symmetric monoidal weakly closed category. We also show how to view general stochastic processes using functor categories and demonstrate the Kalman filter as an archetype for the hidden Markov model.

The parameter space for any fixed architecture of feedforward ReLU neural networks serves as a proxy during training for the associated class of functions - but how faithful is this representation? It is known that many different parameter settings can determine the same function. Moreover, the degree of this redundancy is inhomogeneous: for some networks, the only symmetries are permutation of neurons in a layer and positive scaling of parameters at a neuron, while other networks admit additional hidden symmetries. In this work, we prove that, for any network architecture where no layer is narrower than the input, there exist parameter settings with no hidden symmetries. We also describe a number of mechanisms through which hidden symmetries can arise, and empirically approximate the functional dimension of different network architectures at initialization. These experiments indicate that the probability that a network has no hidden symmetries decreases towards 0 as depth increases, while increasing towards 1 as width and input dimension increase.

Generalized Category Discovery (GCD) is an intriguing open-world problem that has garnered increasing attention. Given a dataset that includes both labelled and unlabelled images, GCD aims to categorize all images in the unlabelled subset, regardless of whether they belong to known or unknown classes. In GCD, the common practice typically involves applying a spherical projection operator at the end of the self-supervised pretrained backbone, operating within Euclidean or spherical space. However, both of these spaces have been shown to be suboptimal for encoding samples that possesses hierarchical structures. In contrast, hyperbolic space exhibits exponential volume growth relative to radius, making it inherently strong at capturing the hierarchical structure of samples from both seen and unseen categories. Therefore, we propose to tackle the category discovery challenge in the hyperbolic space. We introduce HypCD, a simple Hyperbolic framework for learning hierarchy-aware representations and classifiers for generalized Category Discovery. HypCD first transforms the Euclidean embedding space of the backbone network into hyperbolic space, facilitating subsequent representation and classification learning by considering both hyperbolic distance and the angle between samples. This approach is particularly helpful for knowledge transfer from known to unknown categories in GCD. We thoroughly evaluate HypCD on public GCD benchmarks, by applying it to various baseline and state-of-the-art methods, consistently achieving significant improvements.

In this study, we delve into the generation of high-resolution images from pre-trained diffusion models, addressing persistent challenges, such as repetitive patterns and structural distortions, that emerge when models are applied beyond their trained resolutions. To address this issue, we introduce an innovative, training-free approach FouriScale from the perspective of frequency domain analysis. We replace the original convolutional layers in pre-trained diffusion models by incorporating a dilation technique along with a low-pass operation, intending to achieve structural consistency and scale consistency across resolutions, respectively. Further enhanced by a padding-then-crop strategy, our method can flexibly handle text-to-image generation of various aspect ratios. By using the FouriScale as guidance, our method successfully balances the structural integrity and fidelity of generated images, achieving an astonishing capacity of arbitrary-size, high-resolution, and high-quality generation. With its simplicity and compatibility, our method can provide valuable insights for future explorations into the synthesis of ultra-high-resolution images. The code will be released at https://github.com/LeonHLJ/FouriScale.

Objective: Electroencephalography signals are recorded as a multidimensional dataset. We propose a new framework based on the augmented covariance extracted from an autoregressive model to improve motor imagery classification. Methods: From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to extend the standard approach using these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. Results: The augmented covariance matrix performed noticeably better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. Conclusion: The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information, incorporating nonlinear components of the signal through an embedding procedure, which allows the leveraging of dynamical systems algorithms. Significance: These results extend the concepts and the results of the Riemannian distance based classification algorithm.

Recently the author presented a new approach to solving the coefficient problems for various classes of holomorphic functions f(z) = sumlimits_0^infty c_n z^n, not necessarily univalent. This approach is based on lifting the given polynomial coefficient functionals J(f) = J(c_{m_1}, dots, c_{m_s}), 2 < c_{m_1} < dots < c_{m_s} < infty, onto the Bers fiber space over universal Teichmuller space and applying the analytic and geometric features of Teichm\"{u}ller spaces, especially the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces. In this paper, we extend this approach to more general classes of functions. In particular, this provides a strengthening of de Branges' theorem solving the Bieberbach conjecture.

Previous research has shown that the principal singular vectors of a pre-trained model's weight matrices capture critical knowledge. In contrast, those associated with small singular values may contain noise or less reliable information. As a result, the LoRA-based parameter-efficient fine-tuning (PEFT) approach, which does not constrain the use of the spectral space, may not be effective for tasks that demand high representation capacity. In this study, we enhance existing PEFT techniques by incorporating the spectral information of pre-trained weight matrices into the fine-tuning process. We investigate spectral adaptation strategies with a particular focus on the additive adjustment of top singular vectors. This is accomplished by applying singular value decomposition (SVD) to the pre-trained weight matrices and restricting the fine-tuning within the top spectral space. Extensive speaker verification experiments on VoxCeleb1 and CN-Celeb1 demonstrate enhanced tuning performance with the proposed approach. Code is released at https://github.com/lizhepolyu/SpectralFT.

The concept class of low-degree polynomial threshold functions (PTFs) plays a fundamental role in machine learning. In this paper, we study PAC learning of K-sparse degree-d PTFs on R^n, where any such concept depends only on K out of n attributes of the input. Our main contribution is a new algorithm that runs in time ({nd}/{epsilon})^{O(d)} and under the Gaussian marginal distribution, PAC learns the class up to error rate epsilon with O(K^{4d}{epsilon^{2d}} cdot log^{5d} n) samples even when an eta leq O(epsilon^d) fraction of them are corrupted by the nasty noise of Bshouty et al. (2002), possibly the strongest corruption model. Prior to this work, attribute-efficient robust algorithms are established only for the special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a structural result that translates the attribute sparsity to a sparsity pattern of the Chow vector under the basis of Hermite polynomials, and 2) a novel attribute-efficient robust Chow vector estimation algorithm which uses exclusively a restricted Frobenius norm to either certify a good approximation or to validate a sparsity-induced degree-2d polynomial as a filter to detect corrupted samples.

Feature representation in the form of spatio-spectral decomposition is one of the robust techniques adopted in automatic handwritten character recognition systems. In this regard, we propose a new image representation approach for unconstrained handwritten alphanumeric characters using sparse concept coded Tetrolets. Tetrolets, which does not use fixed dyadic square blocks for spectral decomposition like conventional wavelets, preserve the localized variations in handwritings by adopting tetrominoes those capture the shape geometry. The sparse concept coding of low entropy Tetrolet representation is found to extract the important hidden information (concept) for superior pattern discrimination. Large scale experimentation using ten databases in six different scripts (Bangla, Devanagari, Odia, English, Arabic and Telugu) has been performed. The proposed feature representation along with standard classifiers such as random forest, support vector machine (SVM), nearest neighbor and modified quadratic discriminant function (MQDF) is found to achieve state-of-the-art recognition performance in all the databases, viz. 99.40% (MNIST); 98.72% and 93.24% (IITBBS); 99.38% and 99.22% (ISI Kolkata). The proposed OCR system is shown to perform better than other sparse based techniques such as PCA, SparsePCA and SparseLDA, as well as better than existing transforms (Wavelet, Slantlet and Stockwell).

Time series analysis is of immense importance in extensive applications, such as weather forecasting, anomaly detection, and action recognition. This paper focuses on temporal variation modeling, which is the common key problem of extensive analysis tasks. Previous methods attempt to accomplish this directly from the 1D time series, which is extremely challenging due to the intricate temporal patterns. Based on the observation of multi-periodicity in time series, we ravel out the complex temporal variations into the multiple intraperiod- and interperiod-variations. To tackle the limitations of 1D time series in representation capability, we extend the analysis of temporal variations into the 2D space by transforming the 1D time series into a set of 2D tensors based on multiple periods. This transformation can embed the intraperiod- and interperiod-variations into the columns and rows of the 2D tensors respectively, making the 2D-variations to be easily modeled by 2D kernels. Technically, we propose the TimesNet with TimesBlock as a task-general backbone for time series analysis. TimesBlock can discover the multi-periodicity adaptively and extract the complex temporal variations from transformed 2D tensors by a parameter-efficient inception block. Our proposed TimesNet achieves consistent state-of-the-art in five mainstream time series analysis tasks, including short- and long-term forecasting, imputation, classification, and anomaly detection. Code is available at this repository: https://github.com/thuml/TimesNet.

The kernel thinning (KT) algorithm of Dwivedi and Mackey (2021) compresses a probability distribution more effectively than independent sampling by targeting a reproducing kernel Hilbert space (RKHS) and leveraging a less smooth square-root kernel. Here we provide four improvements. First, we show that KT applied directly to the target RKHS yields tighter, dimension-free guarantees for any kernel, any distribution, and any fixed function in the RKHS. Second, we show that, for analytic kernels like Gaussian, inverse multiquadric, and sinc, target KT admits maximum mean discrepancy (MMD) guarantees comparable to or better than those of square-root KT without making explicit use of a square-root kernel. Third, we prove that KT with a fractional power kernel yields better-than-Monte-Carlo MMD guarantees for non-smooth kernels, like Laplace and Mat\'ern, that do not have square-roots. Fourth, we establish that KT applied to a sum of the target and power kernels (a procedure we call KT+) simultaneously inherits the improved MMD guarantees of power KT and the tighter individual function guarantees of target KT. In our experiments with target KT and KT+, we witness significant improvements in integration error even in 100 dimensions and when compressing challenging differential equation posteriors.

This paper presents a data-driven, task-specific paradigm for experimental design, to shorten acquisition time, reduce costs, and accelerate the deployment of imaging devices. Current approaches in experimental design focus on model-parameter estimation and require specification of a particular model, whereas in imaging, other tasks may drive the design. Furthermore, such approaches often lead to intractable optimization problems in real-world imaging applications. Here we present a new paradigm for experimental design that simultaneously optimizes the design (set of image channels) and trains a machine-learning model to execute a user-specified image-analysis task. The approach obtains data densely-sampled over the measurement space (many image channels) for a small number of acquisitions, then identifies a subset of channels of prespecified size that best supports the task. We propose a method: TADRED for TAsk-DRiven Experimental Design in imaging, to identify the most informative channel-subset whilst simultaneously training a network to execute the task given the subset. Experiments demonstrate the potential of TADRED in diverse imaging applications: several clinically-relevant tasks in magnetic resonance imaging; and remote sensing and physiological applications of hyperspectral imaging. Results show substantial improvement over classical experimental design, two recent application-specific methods within the new paradigm, and state-of-the-art approaches in supervised feature selection. We anticipate further applications of our approach. Code is available: https://github.com/sbb-gh/experimental-design-multichannel

Visualizing multiple time series presents fundamental tradeoffs between scalability and visual clarity. Time series capture the behavior of many large-scale real-world processes, from stock market trends to urban activities. Users often gain insights by visualizing them as line charts, juxtaposing or superposing multiple time series to compare them and identify trends and patterns. However, existing representations struggle with scalability: when covering long time spans, leading to visual clutter from too many small multiples or overlapping lines. We propose TiVy, a new algorithm that summarizes time series using sequential patterns. It transforms the series into a set of symbolic sequences based on subsequence visual similarity using Dynamic Time Warping (DTW), then constructs a disjoint grouping of similar subsequences based on the frequent sequential patterns. The grouping result, a visual summary of time series, provides uncluttered superposition with fewer small multiples. Unlike common clustering techniques, TiVy extracts similar subsequences (of varying lengths) aligned in time. We also present an interactive time series visualization that renders large-scale time series in real-time. Our experimental evaluation shows that our algorithm (1) extracts clear and accurate patterns when visualizing time series data, (2) achieves a significant speed-up (1000X) compared to a straightforward DTW clustering. We also demonstrate the efficiency of our approach to explore hidden structures in massive time series data in two usage scenarios.

MRI super-resolution (SR) and denoising tasks are fundamental challenges in the field of deep learning, which have traditionally been treated as distinct tasks with separate paired training data. In this paper, we propose an innovative method that addresses both tasks simultaneously using a single deep learning model, eliminating the need for explicitly paired noisy and clean images during training. Our proposed model is primarily trained for SR, but also exhibits remarkable noise-cleaning capabilities in the super-resolved images. Instead of conventional approaches that introduce frequency-related operations into the generative process, our novel approach involves the use of a GAN model guided by a frequency-informed discriminator. To achieve this, we harness the power of the 3D Discrete Wavelet Transform (DWT) operation as a frequency constraint within the GAN framework for the SR task on magnetic resonance imaging (MRI) data. Specifically, our contributions include: 1) a 3D generator based on residual-in-residual connected blocks; 2) the integration of the 3D DWT with 1times 1 convolution into a DWT+conv unit within a 3D Unet for the discriminator; 3) the use of the trained model for high-quality image SR, accompanied by an intrinsic denoising process. We dub the model "Denoising Induced Super-resolution GAN (DISGAN)" due to its dual effects of SR image generation and simultaneous denoising. Departing from the traditional approach of training SR and denoising tasks as separate models, our proposed DISGAN is trained only on the SR task, but also achieves exceptional performance in denoising. The model is trained on 3D MRI data from dozens of subjects from the Human Connectome Project (HCP) and further evaluated on previously unseen MRI data from subjects with brain tumours and epilepsy to assess its denoising and SR performance.

Autoregressive models have achieved impressive results over a wide range of domains in terms of generation quality and downstream task performance. In the continuous domain, a key factor behind this success is the usage of quantized latent spaces (e.g., obtained via VQ-VAE autoencoders), which allow for dimensionality reduction and faster inference times. However, using existing pre-trained models to perform new non-trivial tasks is difficult since it requires additional fine-tuning or extensive training to elicit prompting. This paper introduces LASS as a way to perform vector-quantized Latent Autoregressive Source Separation (i.e., de-mixing an input signal into its constituent sources) without requiring additional gradient-based optimization or modifications of existing models. Our separation method relies on the Bayesian formulation in which the autoregressive models are the priors, and a discrete (non-parametric) likelihood function is constructed by performing frequency counts over latent sums of addend tokens. We test our method on images and audio with several sampling strategies (e.g., ancestral, beam search) showing competitive results with existing approaches in terms of separation quality while offering at the same time significant speedups in terms of inference time and scalability to higher dimensional data.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators between infinite dimensional spaces, leading to discretization invariant representations that scale independently of the sample grid resolution. Here we present Variational Autoencoding Neural Operators (VANO), a general strategy for making a large class of operator learning architectures act as variational autoencoders. For this purpose, we provide a novel rigorous mathematical formulation of the variational objective in function spaces for training. VANO first maps an input function to a distribution over a latent space using a parametric encoder and then decodes a sample from the latent distribution to reconstruct the input, as in classic variational autoencoders. We test VANO with different model set-ups and architecture choices for a variety of benchmarks. We start from a simple Gaussian random field where we can analytically track what the model learns and progressively transition to more challenging benchmarks including modeling phase separation in Cahn-Hilliard systems and real world satellite data for measuring Earth surface deformation.

In recent years, Printed Circuit Boards (PCB) have become the backbone of a large number of consumer electronic devices leading to a surge in their production. This has made it imperative to employ automatic inspection systems to identify manufacturing defects in PCB before they are installed in the respective systems. An important task in this regard is the classification of defects as either true or pseudo defects, which decides if the PCB is to be re-manufactured or not. This work proposes a novel approach to detect most common defects in the PCBs. The problem has been approached by employing highly discriminative features based on multi-scale wavelet transform, which are further boosted by using a kernalized version of the support vector machines (SVM). A real world printed circuit board dataset has been used for quantitative analysis. Experimental results demonstrated the efficacy of the proposed method.

Image classifiers are known to be difficult to interpret and therefore require explanation methods to understand their decisions. We present ShearletX, a novel mask explanation method for image classifiers based on the shearlet transform -- a multiscale directional image representation. Current mask explanation methods are regularized by smoothness constraints that protect against undesirable fine-grained explanation artifacts. However, the smoothness of a mask limits its ability to separate fine-detail patterns, that are relevant for the classifier, from nearby nuisance patterns, that do not affect the classifier. ShearletX solves this problem by avoiding smoothness regularization all together, replacing it by shearlet sparsity constraints. The resulting explanations consist of a few edges, textures, and smooth parts of the original image, that are the most relevant for the decision of the classifier. To support our method, we propose a mathematical definition for explanation artifacts and an information theoretic score to evaluate the quality of mask explanations. We demonstrate the superiority of ShearletX over previous mask based explanation methods using these new metrics, and present exemplary situations where separating fine-detail patterns allows explaining phenomena that were not explainable before.

Deep sparse networks are widely investigated as a neural network architecture for prediction tasks with high-dimensional sparse features, with which feature interaction selection is a critical component. While previous methods primarily focus on how to search feature interaction in a coarse-grained space, less attention has been given to a finer granularity. In this work, we introduce a hybrid-grained feature interaction selection approach that targets both feature field and feature value for deep sparse networks. To explore such expansive space, we propose a decomposed space which is calculated on the fly. We then develop a selection algorithm called OptFeature, which efficiently selects the feature interaction from both the feature field and the feature value simultaneously. Results from experiments on three large real-world benchmark datasets demonstrate that OptFeature performs well in terms of accuracy and efficiency. Additional studies support the feasibility of our method.

Pre-trained foundation models (FMs) have shown exceptional performance in univariate time series forecasting tasks. However, several practical challenges persist, including managing intricate dependencies among features and quantifying uncertainty in predictions. This study aims to tackle these critical limitations by introducing adapters; feature-space transformations that facilitate the effective use of pre-trained univariate time series FMs for multivariate tasks. Adapters operate by projecting multivariate inputs into a suitable latent space and applying the FM independently to each dimension. Inspired by the literature on representation learning and partially stochastic Bayesian neural networks, we present a range of adapters and optimization/inference strategies. Experiments conducted on both synthetic and real-world datasets confirm the efficacy of adapters, demonstrating substantial enhancements in forecasting accuracy and uncertainty quantification compared to baseline methods. Our framework, AdaPTS, positions adapters as a modular, scalable, and effective solution for leveraging time series FMs in multivariate contexts, thereby promoting their wider adoption in real-world applications. We release the code at https://github.com/abenechehab/AdaPTS.

This paper is the second in the series Commutative Scaling of Width and Depth (WD) about commutativity of infinite width and depth limits in deep neural networks. Our aim is to understand the behaviour of neural functions (functions that depend on a neural network model) as width and depth go to infinity (in some sense), and eventually identify settings under which commutativity holds, i.e. the neural function tends to the same limit no matter how width and depth limits are taken. In this paper, we formally introduce and define the commutativity framework, and discuss its implications on neural network design and scaling. We study commutativity for the neural covariance kernel which reflects how network layers separate data. Our findings extend previous results established in [55] by showing that taking the width and depth to infinity in a deep neural network with skip connections, when branches are suitably scaled to avoid exploding behaviour, result in the same covariance structure no matter how that limit is taken. This has a number of theoretical and practical implications that we discuss in the paper. The proof techniques in this paper are novel and rely on tools that are more accessible to readers who are not familiar with stochastic calculus (used in the proofs of WD(I))).

Accurate 3D cardiac reconstruction from cine magnetic resonance imaging (cMRI) is crucial for improved cardiovascular disease diagnosis and understanding of the heart's motion. However, current cardiac MRI-based reconstruction technology used in clinical settings is 2D with limited through-plane resolution, resulting in low-quality reconstructed cardiac volumes. To better reconstruct 3D cardiac volumes from sparse 2D image stacks, we propose a morphology-guided diffusion model for 3D cardiac volume reconstruction, DMCVR, that synthesizes high-resolution 2D images and corresponding 3D reconstructed volumes. Our method outperforms previous approaches by conditioning the cardiac morphology on the generative model, eliminating the time-consuming iterative optimization process of the latent code, and improving generation quality. The learned latent spaces provide global semantics, local cardiac morphology and details of each 2D cMRI slice with highly interpretable value to reconstruct 3D cardiac shape. Our experiments show that DMCVR is highly effective in several aspects, such as 2D generation and 3D reconstruction performance. With DMCVR, we can produce high-resolution 3D cardiac MRI reconstructions, surpassing current techniques. Our proposed framework has great potential for improving the accuracy of cardiac disease diagnosis and treatment planning. Code can be accessed at https://github.com/hexiaoxiao-cs/DMCVR.

Hyperbolic neural networks have shown great potential for modeling complex data. However, existing hyperbolic networks are not completely hyperbolic, as they encode features in a hyperbolic space yet formalize most of their operations in the tangent space (a Euclidean subspace) at the origin of the hyperbolic space. This hybrid method greatly limits the modeling ability of networks. In this paper, we propose a fully hyperbolic framework to build hyperbolic networks based on the Lorentz model by adapting the Lorentz transformations (including boost and rotation) to formalize essential operations of neural networks. Moreover, we also prove that linear transformation in tangent spaces used by existing hyperbolic networks is a relaxation of the Lorentz rotation and does not include the boost, implicitly limiting the capabilities of existing hyperbolic networks. The experimental results on four NLP tasks show that our method has better performance for building both shallow and deep networks. Our code will be released to facilitate follow-up research.

Foundation models have triggered a paradigm shift in computer vision and are increasingly being adopted in remote sensing, particularly for multispectral imagery. Yet, their potential in hyperspectral imaging (HSI) remains untapped due to the absence of comprehensive and globally representative hyperspectral datasets. To close this gap, we introduce SpectralEarth, a large-scale multi-temporal dataset designed to pretrain hyperspectral foundation models leveraging data from the Environmental Mapping and Analysis Program (EnMAP). SpectralEarth comprises 538,974 image patches covering 415,153 unique locations from more than 11,636 globally distributed EnMAP scenes spanning two years of archive. Additionally, 17.5% of these locations include multiple timestamps, enabling multi-temporal HSI analysis. Utilizing state-of-the-art self-supervised learning (SSL) algorithms, we pretrain a series of foundation models on SpectralEarth. We integrate a spectral adapter into classical vision backbones to accommodate the unique characteristics of HSI. In tandem, we construct four downstream datasets for land-cover and crop-type mapping, providing benchmarks for model evaluation. Experimental results support the versatility of our models, showcasing their generalizability across different tasks and sensors. We also highlight computational efficiency during model fine-tuning. The dataset, models, and source code will be made publicly available.

This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric. The flag manifold is a mathematical space consisting of flags, which are sequences of nested subspaces of a vector space that increase in dimension. The flag manifold is a superset of a wide range of known matrix spaces, including Stiefel and Grassmanians, making it a general object that is useful in a wide variety computer vision problems. To tackle the challenge of computing first order flag statistics, we first transform the problem into one that involves auxiliary variables constrained to the Stiefel manifold. The Stiefel manifold is a space of orthogonal frames, and leveraging the numerical stability and efficiency of Stiefel-manifold optimization enables us to compute the flag-mean effectively. Through a series of experiments, we show the competence of our method in Grassmann and rotation averaging, as well as principal component analysis. We release our source code under https://github.com/nmank/FlagAveraging.

In autoregressive (AR) image generation, visual tokenizers compress images into compact discrete latent tokens, enabling efficient training of downstream autoregressive models for visual generation via next-token prediction. While scaling visual tokenizers improves image reconstruction quality, it often degrades downstream generation quality -- a challenge not adequately addressed in existing literature. To address this, we introduce GigaTok, the first approach to simultaneously improve image reconstruction, generation, and representation learning when scaling visual tokenizers. We identify the growing complexity of latent space as the key factor behind the reconstruction vs. generation dilemma. To mitigate this, we propose semantic regularization, which aligns tokenizer features with semantically consistent features from a pre-trained visual encoder. This constraint prevents excessive latent space complexity during scaling, yielding consistent improvements in both reconstruction and downstream autoregressive generation. Building on semantic regularization, we explore three key practices for scaling tokenizers:(1) using 1D tokenizers for better scalability, (2) prioritizing decoder scaling when expanding both encoder and decoder, and (3) employing entropy loss to stabilize training for billion-scale tokenizers. By scaling to 3 space billion parameters, GigaTok achieves state-of-the-art performance in reconstruction, downstream AR generation, and downstream AR representation quality.

We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting G-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

Latent diffusion models have become the popular choice for scaling up diffusion models for high resolution image synthesis. Compared to pixel-space models that are trained end-to-end, latent models are perceived to be more efficient and to produce higher image quality at high resolution. Here we challenge these notions, and show that pixel-space models can in fact be very competitive to latent approaches both in quality and efficiency, achieving 1.5 FID on ImageNet512 and new SOTA results on ImageNet128 and ImageNet256. We present a simple recipe for scaling end-to-end pixel-space diffusion models to high resolutions. 1: Use the sigmoid loss (Kingma & Gao, 2023) with our prescribed hyper-parameters. 2: Use our simplified memory-efficient architecture with fewer skip-connections. 3: Scale the model to favor processing the image at high resolution with fewer parameters, rather than using more parameters but at a lower resolution. When combining these three steps with recently proposed tricks like guidance intervals, we obtain a family of pixel-space diffusion models we call Simple Diffusion v2 (SiD2).

Advanced interpretation of hyperspectral remote sensing images benefits many precise Earth observation tasks. Recently, visual foundation models have promoted the remote sensing interpretation but concentrating on RGB and multispectral images. Due to the varied hyperspectral channels,existing foundation models would face image-by-image tuning situation, imposing great pressure on hardware and time resources. In this paper, we propose a tuning-free hyperspectral foundation model called HyperFree, by adapting the existing visual prompt engineering. To process varied channel numbers, we design a learned weight dictionary covering full-spectrum from 0.4 sim 2.5 , mum, supporting to build the embedding layer dynamically. To make the prompt design more tractable, HyperFree can generate multiple semantic-aware masks for one prompt by treating feature distance as semantic-similarity. After pre-training HyperFree on constructed large-scale high-resolution hyperspectral images, HyperFree (1 prompt) has shown comparable results with specialized models (5 shots) on 5 tasks and 11 datasets.Code and dataset are accessible at https://rsidea.whu.edu.cn/hyperfree.htm.

Large, pretrained models are commonly finetuned with imagery that is heavily augmented to mimic different conditions and scales, with the resulting models used for various tasks with imagery from a range of spatial scales. Such models overlook scale-specific information in the data for scale-dependent domains, such as remote sensing. In this paper, we present Scale-MAE, a pretraining method that explicitly learns relationships between data at different, known scales throughout the pretraining process. Scale-MAE pretrains a network by masking an input image at a known input scale, where the area of the Earth covered by the image determines the scale of the ViT positional encoding, not the image resolution. Scale-MAE encodes the masked image with a standard ViT backbone, and then decodes the masked image through a bandpass filter to reconstruct low/high frequency images at lower/higher scales. We find that tasking the network with reconstructing both low/high frequency images leads to robust multiscale representations for remote sensing imagery. Scale-MAE achieves an average of a 2.4 - 5.6% non-parametric kNN classification improvement across eight remote sensing datasets compared to current state-of-the-art and obtains a 0.9 mIoU to 1.7 mIoU improvement on the SpaceNet building segmentation transfer task for a range of evaluation scales.

In the realms of computer vision, it is evident that deep neural networks perform better in a supervised setting with a large amount of labeled data. The representations learned with supervision are not only of high quality but also helps the model in enhancing its accuracy. However, the collection and annotation of a large dataset are costly and time-consuming. To avoid the same, there has been a lot of research going on in the field of unsupervised visual representation learning especially in a self-supervised setting. Amongst the recent advancements in self-supervised methods for visual recognition, in SimCLR Chen et al. shows that good quality representations can indeed be learned without explicit supervision. In SimCLR, the authors maximize the similarity of augmentations of the same image and minimize the similarity of augmentations of different images. A linear classifier trained with the representations learned using this approach yields 76.5% top-1 accuracy on the ImageNet ILSVRC-2012 dataset. In this work, we propose that, with the normalized temperature-scaled cross-entropy (NT-Xent) loss function (as used in SimCLR), it is beneficial to not have images of the same category in the same batch. In an unsupervised setting, the information of images pertaining to the same category is missing. We use the latent space representation of a denoising autoencoder trained on the unlabeled dataset and cluster them with k-means to obtain pseudo labels. With this apriori information we batch images, where no two images from the same category are to be found. We report comparable performance enhancements on the CIFAR10 dataset and a subset of the ImageNet dataset. We refer to our method as G-SimCLR.

Hyperspectral pansharpening consists of fusing a high-resolution panchromatic band and a low-resolution hyperspectral image to obtain a new image with high resolution in both the spatial and spectral domains. These remote sensing products are valuable for a wide range of applications, driving ever growing research efforts. Nonetheless, results still do not meet application demands. In part, this comes from the technical complexity of the task: compared to multispectral pansharpening, many more bands are involved, in a spectral range only partially covered by the panchromatic component and with overwhelming noise. However, another major limiting factor is the absence of a comprehensive framework for the rapid development and accurate evaluation of new methods. This paper attempts to address this issue. We started by designing a dataset large and diverse enough to allow reliable training (for data-driven methods) and testing of new methods. Then, we selected a set of state-of-the-art methods, following different approaches, characterized by promising performance, and reimplemented them in a single PyTorch framework. Finally, we carried out a critical comparative analysis of all methods, using the most accredited quality indicators. The analysis highlights the main limitations of current solutions in terms of spectral/spatial quality and computational efficiency, and suggests promising research directions. To ensure full reproducibility of the results and support future research, the framework (including codes, evaluation procedures and links to the dataset) is shared on https://github.com/matciotola/hyperspectral_pansharpening_toolbox, as a single Python-based reference benchmark toolbox.

State-space models (SSMs) have recently emerged as a framework for learning long-range sequence tasks. An example is the structured state-space sequence (S4) layer, which uses the diagonal-plus-low-rank structure of the HiPPO initialization framework. However, the complicated structure of the S4 layer poses challenges; and, in an effort to address these challenges, models such as S4D and S5 have considered a purely diagonal structure. This choice simplifies the implementation, improves computational efficiency, and allows channel communication. However, diagonalizing the HiPPO framework is itself an ill-posed problem. In this paper, we propose a general solution for this and related ill-posed diagonalization problems in machine learning. We introduce a generic, backward-stable "perturb-then-diagonalize" (PTD) methodology, which is based on the pseudospectral theory of non-normal operators, and which may be interpreted as the approximate diagonalization of the non-normal matrices defining SSMs. Based on this, we introduce the S4-PTD and S5-PTD models. Through theoretical analysis of the transfer functions of different initialization schemes, we demonstrate that the S4-PTD/S5-PTD initialization strongly converges to the HiPPO framework, while the S4D/S5 initialization only achieves weak convergences. As a result, our new models show resilience to Fourier-mode noise-perturbed inputs, a crucial property not achieved by the S4D/S5 models. In addition to improved robustness, our S5-PTD model averages 87.6% accuracy on the Long-Range Arena benchmark, demonstrating that the PTD methodology helps to improve the accuracy of deep learning models.

Imposing orthogonality on the layers of neural networks is known to facilitate the learning by limiting the exploding/vanishing of the gradient; decorrelate the features; improve the robustness. This paper studies the theoretical properties of orthogonal convolutional layers.We establish necessary and sufficient conditions on the layer architecture guaranteeing the existence of an orthogonal convolutional transform. The conditions prove that orthogonal convolutional transforms exist for almost all architectures used in practice for 'circular' padding.We also exhibit limitations with 'valid' boundary conditions and 'same' boundary conditions with zero-padding.Recently, a regularization term imposing the orthogonality of convolutional layers has been proposed, and impressive empirical results have been obtained in different applications (Wang et al. 2020).The second motivation of the present paper is to specify the theory behind this.We make the link between this regularization term and orthogonality measures. In doing so, we show that this regularization strategy is stable with respect to numerical and optimization errors and that, in the presence of small errors and when the size of the signal/image is large, the convolutional layers remain close to isometric.The theoretical results are confirmed with experiments and the landscape of the regularization term is studied. Experiments on real data sets show that when orthogonality is used to enforce robustness, the parameter multiplying the regularization termcan be used to tune a tradeoff between accuracy and orthogonality, for the benefit of both accuracy and robustness.Altogether, the study guarantees that the regularization proposed in Wang et al. (2020) is an efficient, flexible and stable numerical strategy to learn orthogonal convolutional layers.

Which functions can be used as activations in deep neural networks? This article explores families of functions based on orthonormal bases, including the Hermite polynomial basis and the Fourier trigonometric basis, as well as a basis resulting from the tropicalization of a polynomial basis. Our study shows that, through simple variance-preserving initialization and without additional clamping mechanisms, these activations can successfully be used to train deep models, such as GPT-2 for next-token prediction on OpenWebText and ConvNeXt for image classification on ImageNet. Our work addresses the issue of exploding and vanishing activations and gradients, particularly prevalent with polynomial activations, and opens the door for improving the efficiency of large-scale learning tasks. Furthermore, our approach provides insight into the structure of neural networks, revealing that networks with polynomial activations can be interpreted as multivariate polynomial mappings. Finally, using Hermite interpolation, we show that our activations can closely approximate classical ones in pre-trained models by matching both the function and its derivative, making them especially useful for fine-tuning tasks. These activations are available in the torchortho library, which can be accessed via: https://github.com/K-H-Ismail/torchortho.

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the i-th neuron in a nonlinear operator layer is defined by mathcal O_i(u) = sigmaleft( sum_j mathcal W_{ij} u + mathcal B_{ij}right). Here, mathcal W_{ij} denotes the bounded linear operator connecting j-th input neuron to i-th output neuron, and the bias mathcal B_{ij} takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

Strategies for the generation of periodic discrete structures with identical two-point correlation are developed. Starting from a pair of root structures, which are not related by translation, phase inversion or axis reflections, child structures of arbitrary resolution (i.e., pixel or voxel numbers) and number of phases (i.e., material phases/species) can be generated by means of trivial embedding based phase extension, application of kernels and/or phase coalescence, such that the generated structures inherit the two-point-correlation equivalence. Proofs of the inheritance property are provided by means of the Discrete Fourier Transform theory. A Python 3 implementation of the results is offered by the authors through the Github repository https://github.com/DataAnalyticsEngineering/EQ2PC in order to make the provided results reproducible and useful for all interested readers. Examples for the generation of structures are demonstrated, together with applications in the homogenization theory of periodic media.

Existing works have extensively studied adversarial examples, which are minimal perturbations that can mislead the output of deep neural networks (DNNs) while remaining imperceptible to humans. However, in this work, we reveal the existence of a harmless perturbation space, in which perturbations drawn from this space, regardless of their magnitudes, leave the network output unchanged when applied to inputs. Essentially, the harmless perturbation space emerges from the usage of non-injective functions (linear or non-linear layers) within DNNs, enabling multiple distinct inputs to be mapped to the same output. For linear layers with input dimensions exceeding output dimensions, any linear combination of the orthogonal bases of the nullspace of the parameter consistently yields no change in their output. For non-linear layers, the harmless perturbation space may expand, depending on the properties of the layers and input samples. Inspired by this property of DNNs, we solve for a family of general perturbation spaces that are redundant for the DNN's decision, and can be used to hide sensitive data and serve as a means of model identification. Our work highlights the distinctive robustness of DNNs (i.e., consistency under large magnitude perturbations) in contrast to adversarial examples (vulnerability for small imperceptible noises).

In speech enhancement and source separation, signal-to-noise ratio is a ubiquitous objective measure of denoising/separation quality. A decade ago, the BSS_eval toolkit was developed to give researchers worldwide a way to evaluate the quality of their algorithms in a simple, fair, and hopefully insightful way: it attempted to account for channel variations, and to not only evaluate the total distortion in the estimated signal but also split it in terms of various factors such as remaining interference, newly added artifacts, and channel errors. In recent years, hundreds of papers have been relying on this toolkit to evaluate their proposed methods and compare them to previous works, often arguing that differences on the order of 0.1 dB proved the effectiveness of a method over others. We argue here that the signal-to-distortion ratio (SDR) implemented in the BSS_eval toolkit has generally been improperly used and abused, especially in the case of single-channel separation, resulting in misleading results. We propose to use a slightly modified definition, resulting in a simpler, more robust measure, called scale-invariant SDR (SI-SDR). We present various examples of critical failure of the original SDR that SI-SDR overcomes.

Hyperbolic geometry is gaining traction in machine learning for its effectiveness at capturing hierarchical structures in real-world data. Hyperbolic spaces, where neighborhoods grow exponentially, offer substantial advantages and consistently deliver state-of-the-art results across diverse applications. However, hyperbolic classifiers often grapple with computational challenges. Methods reliant on Riemannian optimization frequently exhibit sluggishness, stemming from the increased computational demands of operations on Riemannian manifolds. In response to these challenges, we present hyperDT, a novel extension of decision tree algorithms into hyperbolic space. Crucially, hyperDT eliminates the need for computationally intensive Riemannian optimization, numerically unstable exponential and logarithmic maps, or pairwise comparisons between points by leveraging inner products to adapt Euclidean decision tree algorithms to hyperbolic space. Our approach is conceptually straightforward and maintains constant-time decision complexity while mitigating the scalability issues inherent in high-dimensional Euclidean spaces. Building upon hyperDT we introduce hyperRF, a hyperbolic random forest model. Extensive benchmarking across diverse datasets underscores the superior performance of these models, providing a swift, precise, accurate, and user-friendly toolkit for hyperbolic data analysis.

Dictionary learning is a branch of signal processing and machine learning that aims at finding a frame (called dictionary) in which some training data admits a sparse representation. The sparser the representation, the better the dictionary. The resulting dictionary is in general a dense matrix, and its manipulation can be computationally costly both at the learning stage and later in the usage of this dictionary, for tasks such as sparse coding. Dictionary learning is thus limited to relatively small-scale problems. In this paper, inspired by usual fast transforms, we consider a general dictionary structure that allows cheaper manipulation, and propose an algorithm to learn such dictionaries --and their fast implementation-- over training data. The approach is demonstrated experimentally with the factorization of the Hadamard matrix and with synthetic dictionary learning experiments.

We propose PROSE-FD, a zero-shot multimodal PDE foundational model for simultaneous prediction of heterogeneous two-dimensional physical systems related to distinct fluid dynamics settings. These systems include shallow water equations and the Navier-Stokes equations with incompressible and compressible flow, regular and complex geometries, and different buoyancy settings. This work presents a new transformer-based multi-operator learning approach that fuses symbolic information to perform operator-based data prediction, i.e. non-autoregressive. By incorporating multiple modalities in the inputs, the PDE foundation model builds in a pathway for including mathematical descriptions of the physical behavior. We pre-train our foundation model on 6 parametric families of equations collected from 13 datasets, including over 60K trajectories. Our model outperforms popular operator learning, computer vision, and multi-physics models, in benchmark forward prediction tasks. We test our architecture choices with ablation studies.

With the advent of surveys like Euclid and Vera C. Rubin, astrophysicists will have access to both deep, high-resolution images and multiband images. However, these two types are not simultaneously available in any single dataset. It is therefore vital to devise image deconvolution algorithms that exploit the best of both worlds and that can jointly analyze datasets spanning a range of resolutions and wavelengths. In this work we introduce a novel multiband deconvolution technique aimed at improving the resolution of ground-based astronomical images by leveraging higher-resolution space-based observations. The method capitalizes on the fortunate fact that the Rubin r, i, and z bands lie within the Euclid VIS band. The algorithm jointly de-convolves all the data to convert the r-, i-, and z-band Rubin images to the resolution of Euclid by leveraging the correlations between the different bands. We also investigate the performance of deep-learning-based denoising with DRUNet to further improve the results. We illustrate the effectiveness of our method in terms of resolution and morphology recovery, flux preservation, and generalization to different noise levels. This approach extends beyond the specific Euclid-Rubin combination, offering a versatile solution to improving the resolution of ground-based images in multiple photometric bands by jointly using any space-based images with overlapping filters.

Standard methods for detecting discontinuities in conditional means are not applicable to outcomes that are complex, non-Euclidean objects like distributions, networks, or covariance matrices. This article develops a nonparametric test for jumps in conditional means when outcomes lie in a non-Euclidean metric space. Using local Fr\'echet regressionx2014which generalizes standard regression to metric-space valued datax2014the method estimates a mean path on either side of a candidate cutoff, extending existing k-sample tests to a flexible regression setting. Key theoretical contributions include a central limit theorem for the local estimator of the conditional Fr\'echet variance and the asymptotic validity and consistency of the proposed test. Simulations confirm nominal size control and robust power in finite samples. Two applications demonstrate the method's value by revealing effects invisible to scalar-based tests. First, I detect a sharp change in work-from-home compositions at Washington State's income threshold for non-compete enforceability during COVID-19, highlighting remote work's role as a bargaining margin. Second, I find that countries restructure their input-output networks after losing preferential US trade access. These findings underscore that analyzing regression functions within their native metric spaces can reveal structural discontinuities that scalar summaries would miss.

Ultra-high-resolution image synthesis holds significant potential, yet remains an underexplored challenge due to the absence of standardized benchmarks and computational constraints. In this paper, we establish Aesthetic-4K, a meticulously curated dataset containing dedicated training and evaluation subsets specifically designed for comprehensive research on ultra-high-resolution image synthesis. This dataset consists of high-quality 4K images accompanied by descriptive captions generated by GPT-4o. Furthermore, we propose Diffusion-4K, an innovative framework for the direct generation of ultra-high-resolution images. Our approach incorporates the Scale Consistent Variational Auto-Encoder (SC-VAE) and Wavelet-based Latent Fine-tuning (WLF), which are designed for efficient visual token compression and the capture of intricate details in ultra-high-resolution images, thereby facilitating direct training with photorealistic 4K data. This method is applicable to various latent diffusion models and demonstrates its efficacy in synthesizing highly detailed 4K images. Additionally, we propose novel metrics, namely the GLCM Score and Compression Ratio, to assess the texture richness and fine details in local patches, in conjunction with holistic measures such as FID, Aesthetics, and CLIPScore, enabling a thorough and multifaceted evaluation of ultra-high-resolution image synthesis. Consequently, Diffusion-4K achieves impressive performance in ultra-high-resolution image synthesis, particularly when powered by state-of-the-art large-scale diffusion models (eg, Flux-12B). The source code is publicly available at https://github.com/zhang0jhon/diffusion-4k.

Sequential learning paradigms pose challenges for gradient-based deep learning due to difficulties incorporating new data and retaining prior knowledge. While Gaussian processes elegantly tackle these problems, they struggle with scalability and handling rich inputs, such as images. To address these issues, we introduce a technique that converts neural networks from weight space to function space, through a dual parameterization. Our parameterization offers: (i) a way to scale function-space methods to large data sets via sparsification, (ii) retention of prior knowledge when access to past data is limited, and (iii) a mechanism to incorporate new data without retraining. Our experiments demonstrate that we can retain knowledge in continual learning and incorporate new data efficiently. We further show its strengths in uncertainty quantification and guiding exploration in model-based RL. Further information and code is available on the project website.

State space models (SSM) have recently been shown to be very effective as a deep learning layer as a promising alternative to sequence models such as RNNs, CNNs, or Transformers. The first version to show this potential was the S4 model, which is particularly effective on tasks involving long-range dependencies by using a prescribed state matrix called the HiPPO matrix. While this has an interpretable mathematical mechanism for modeling long dependencies, it introduces a custom representation and algorithm that can be difficult to implement. On the other hand, a recent variant of S4 called DSS showed that restricting the state matrix to be fully diagonal can still preserve the performance of the original model when using a specific initialization based on approximating S4's matrix. This work seeks to systematically understand how to parameterize and initialize such diagonal state space models. While it follows from classical results that almost all SSMs have an equivalent diagonal form, we show that the initialization is critical for performance. We explain why DSS works mathematically, by showing that the diagonal restriction of S4's matrix surprisingly recovers the same kernel in the limit of infinite state dimension. We also systematically describe various design choices in parameterizing and computing diagonal SSMs, and perform a controlled empirical study ablating the effects of these choices. Our final model S4D is a simple diagonal version of S4 whose kernel computation requires just 2 lines of code and performs comparably to S4 in almost all settings, with state-of-the-art results for image, audio, and medical time-series domains, and averaging 85\% on the Long Range Arena benchmark.

Neural Radiance Fields (NeRF) has achieved superior performance in novel view synthesis and 3D scene representation, but its practical applications are hindered by slow convergence and reliance on dense training views. To this end, we present DWTNeRF, a unified framework based on Instant-NGP's fast-training hash encoding. It is coupled with regularization terms designed for few-shot NeRF, which operates on sparse training views. Our DWTNeRF additionally includes a novel Discrete Wavelet loss that allows explicit prioritization of low frequencies directly in the training objective, reducing few-shot NeRF's overfitting on high frequencies in earlier training stages. We also introduce a model-based approach, based on multi-head attention, that is compatible with INGP, which are sensitive to architectural changes. On the 3-shot LLFF benchmark, DWTNeRF outperforms Vanilla INGP by 15.07% in PSNR, 24.45% in SSIM and 36.30% in LPIPS. Our approach encourages a re-thinking of current few-shot approaches for fast-converging implicit representations like INGP or 3DGS.

Many learning algorithms have invariances: when their training data is transformed in certain ways, the function they learn transforms in a predictable manner. Here we formalize this notion using concepts from the mathematical field of category theory. The invariances that a supervised learning algorithm possesses are formalized by categories of predictor and target spaces, whose morphisms represent the algorithm's invariances, and an index category whose morphisms represent permutations of the training examples. An invariant learning algorithm is a natural transformation between two functors from the product of these categories to the category of sets, representing training datasets and learned functions respectively. We illustrate the framework by characterizing and contrasting the invariances of linear regression and ridge regression.

We propose Super-resolution Neural Operator (SRNO), a deep operator learning framework that can resolve high-resolution (HR) images at arbitrary scales from the low-resolution (LR) counterparts. Treating the LR-HR image pairs as continuous functions approximated with different grid sizes, SRNO learns the mapping between the corresponding function spaces. From the perspective of approximation theory, SRNO first embeds the LR input into a higher-dimensional latent representation space, trying to capture sufficient basis functions, and then iteratively approximates the implicit image function with a kernel integral mechanism, followed by a final dimensionality reduction step to generate the RGB representation at the target coordinates. The key characteristics distinguishing SRNO from prior continuous SR works are: 1) the kernel integral in each layer is efficiently implemented via the Galerkin-type attention, which possesses non-local properties in the spatial domain and therefore benefits the grid-free continuum; and 2) the multilayer attention architecture allows for the dynamic latent basis update, which is crucial for SR problems to "hallucinate" high-frequency information from the LR image. Experiments show that SRNO outperforms existing continuous SR methods in terms of both accuracy and running time. Our code is at https://github.com/2y7c3/Super-Resolution-Neural-Operator

Unsupervised hashing methods typically aim to preserve the similarity between data points in a feature space by mapping them to binary hash codes. However, these methods often overlook the fact that the similarity between data points in the continuous feature space may not be preserved in the discrete hash code space, due to the limited similarity range of hash codes. The similarity range is bounded by the code length and can lead to a problem known as similarity collapse. That is, the positive and negative pairs of data points become less distinguishable from each other in the hash space. To alleviate this problem, in this paper a novel Similarity Distribution Calibration (SDC) method is introduced. SDC aligns the hash code similarity distribution towards a calibration distribution (e.g., beta distribution) with sufficient spread across the entire similarity range, thus alleviating the similarity collapse problem. Extensive experiments show that our SDC outperforms significantly the state-of-the-art alternatives on coarse category-level and instance-level image retrieval. Code is available at https://github.com/kamwoh/sdc.

We generalize the class vectors found in neural networks to linear subspaces (i.e.~points in the Grassmann manifold) and show that the Grassmann Class Representation (GCR) enables the simultaneous improvement in accuracy and feature transferability. In GCR, each class is a subspace and the logit is defined as the norm of the projection of a feature onto the class subspace. We integrate Riemannian SGD into deep learning frameworks such that class subspaces in a Grassmannian are jointly optimized with the rest model parameters. Compared to the vector form, the representative capability of subspaces is more powerful. We show that on ImageNet-1K, the top-1 error of ResNet50-D, ResNeXt50, Swin-T and Deit3-S are reduced by 5.6%, 4.5%, 3.0% and 3.5%, respectively. Subspaces also provide freedom for features to vary and we observed that the intra-class feature variability grows when the subspace dimension increases. Consequently, we found the quality of GCR features is better for downstream tasks. For ResNet50-D, the average linear transfer accuracy across 6 datasets improves from 77.98% to 79.70% compared to the strong baseline of vanilla softmax. For Swin-T, it improves from 81.5% to 83.4% and for Deit3, it improves from 73.8% to 81.4%. With these encouraging results, we believe that more applications could benefit from the Grassmann class representation. Code is released at https://github.com/innerlee/GCR.

Mamba, a special case of the State Space Model, is gaining popularity as an alternative to template-based deep learning approaches in medical image analysis. While transformers are powerful architectures, they have drawbacks, including quadratic computational complexity and an inability to address long-range dependencies efficiently. This limitation affects the analysis of large and complex datasets in medical imaging, where there are many spatial and temporal relationships. In contrast, Mamba offers benefits that make it well-suited for medical image analysis. It has linear time complexity, which is a significant improvement over transformers. Mamba processes longer sequences without attention mechanisms, enabling faster inference and requiring less memory. Mamba also demonstrates strong performance in merging multimodal data, improving diagnosis accuracy and patient outcomes. The organization of this paper allows readers to appreciate the capabilities of Mamba in medical imaging step by step. We begin by defining core concepts of SSMs and models, including S4, S5, and S6, followed by an exploration of Mamba architectures such as pure Mamba, U-Net variants, and hybrid models with convolutional neural networks, transformers, and Graph Neural Networks. We also cover Mamba optimizations, techniques and adaptations, scanning, datasets, applications, experimental results, and conclude with its challenges and future directions in medical imaging. This review aims to demonstrate the transformative potential of Mamba in overcoming existing barriers within medical imaging while paving the way for innovative advancements in the field. A comprehensive list of Mamba architectures applied in the medical field, reviewed in this work, is available at Github.

Denoising diffusion probabilistic models (DDPMs) employ a sequence of white Gaussian noise samples to generate an image. In analogy with GANs, those noise maps could be considered as the latent code associated with the generated image. However, this native noise space does not possess a convenient structure, and is thus challenging to work with in editing tasks. Here, we propose an alternative latent noise space for DDPM that enables a wide range of editing operations via simple means, and present an inversion method for extracting these edit-friendly noise maps for any given image (real or synthetically generated). As opposed to the native DDPM noise space, the edit-friendly noise maps do not have a standard normal distribution and are not statistically independent across timesteps. However, they allow perfect reconstruction of any desired image, and simple transformations on them translate into meaningful manipulations of the output image (e.g., shifting, color edits). Moreover, in text-conditional models, fixing those noise maps while changing the text prompt, modifies semantics while retaining structure. We illustrate how this property enables text-based editing of real images via the diverse DDPM sampling scheme (in contrast to the popular non-diverse DDIM inversion). We also show how it can be used within existing diffusion-based editing methods to improve their quality and diversity.

The magnitude of a finite metric space has recently emerged as a novel invariant quantity, allowing to measure the effective size of a metric space. Despite encouraging first results demonstrating the descriptive abilities of the magnitude, such as being able to detect the boundary of a metric space, the potential use cases of magnitude remain under-explored. In this work, we investigate the properties of the magnitude on images, an important data modality in many machine learning applications. By endowing each individual images with its own metric space, we are able to define the concept of magnitude on images and analyse the individual contribution of each pixel with the magnitude vector. In particular, we theoretically show that the previously known properties of boundary detection translate to edge detection abilities in images. Furthermore, we demonstrate practical use cases of magnitude for machine learning applications and propose a novel magnitude model that consists of a computationally efficient magnitude computation and a learnable metric. By doing so, we address the computational hurdle that used to make magnitude impractical for many applications and open the way for the adoption of magnitude in machine learning research.

We propose the Signal Dice Similarity Coefficient (SDSC), a structure-aware metric function for time series self-supervised representation learning. Most Self-Supervised Learning (SSL) methods for signals commonly adopt distance-based objectives such as mean squared error (MSE), which are sensitive to amplitude, invariant to waveform polarity, and unbounded in scale. These properties hinder semantic alignment and reduce interpretability. SDSC addresses this by quantifying structural agreement between temporal signals based on the intersection of signed amplitudes, derived from the Dice Similarity Coefficient (DSC).Although SDSC is defined as a structure-aware metric, it can be used as a loss by subtracting from 1 and applying a differentiable approximation of the Heaviside function for gradient-based optimization. A hybrid loss formulation is also proposed to combine SDSC with MSE, improving stability and preserving amplitude where necessary. Experiments on forecasting and classification benchmarks demonstrate that SDSC-based pre-training achieves comparable or improved performance over MSE, particularly in in-domain and low-resource scenarios. The results suggest that structural fidelity in signal representations enhances the semantic representation quality, supporting the consideration of structure-aware metrics as viable alternatives to conventional distance-based methods.

In this paper, we prove representation bottlenecks of a cascaded convolutional decoder network, considering the capacity of representing different frequency components of an input sample. We conduct the discrete Fourier transform on each channel of the feature map in an intermediate layer of the decoder network. Then, we introduce the rule of the forward propagation of such intermediate-layer spectrum maps, which is equivalent to the forward propagation of feature maps through a convolutional layer. Based on this, we find that each frequency component in the spectrum map is forward propagated independently with other frequency components. Furthermore, we prove two bottlenecks in representing feature spectrums. First, we prove that the convolution operation, the zero-padding operation, and a set of other settings all make a convolutional decoder network more likely to weaken high-frequency components. Second, we prove that the upsampling operation generates a feature spectrum, in which strong signals repetitively appears at certain frequencies.

Neural networks embed the geometric structure of a data manifold lying in a high-dimensional space into latent representations. Ideally, the distribution of the data points in the latent space should depend only on the task, the data, the loss, and other architecture-specific constraints. However, factors such as the random weights initialization, training hyperparameters, or other sources of randomness in the training phase may induce incoherent latent spaces that hinder any form of reuse. Nevertheless, we empirically observe that, under the same data and modeling choices, the angles between the encodings within distinct latent spaces do not change. In this work, we propose the latent similarity between each sample and a fixed set of anchors as an alternative data representation, demonstrating that it can enforce the desired invariances without any additional training. We show how neural architectures can leverage these relative representations to guarantee, in practice, invariance to latent isometries and rescalings, effectively enabling latent space communication: from zero-shot model stitching to latent space comparison between diverse settings. We extensively validate the generalization capability of our approach on different datasets, spanning various modalities (images, text, graphs), tasks (e.g., classification, reconstruction) and architectures (e.g., CNNs, GCNs, transformers).

Large-scale 3D generative models require substantial computational resources yet often fall short in capturing fine details and complex geometries at high resolutions. We attribute this limitation to the inefficiency of current representations, which lack the compactness required to model the generative models effectively. To address this, we introduce a novel approach called Wavelet Latent Diffusion, or WaLa, that encodes 3D shapes into wavelet-based, compact latent encodings. Specifically, we compress a 256^3 signed distance field into a 12^3 times 4 latent grid, achieving an impressive 2427x compression ratio with minimal loss of detail. This high level of compression allows our method to efficiently train large-scale generative networks without increasing the inference time. Our models, both conditional and unconditional, contain approximately one billion parameters and successfully generate high-quality 3D shapes at 256^3 resolution. Moreover, WaLa offers rapid inference, producing shapes within two to four seconds depending on the condition, despite the model's scale. We demonstrate state-of-the-art performance across multiple datasets, with significant improvements in generation quality, diversity, and computational efficiency. We open-source our code and, to the best of our knowledge, release the largest pretrained 3D generative models across different modalities.

Iterative algorithms based on thresholding, feedback and null space tuning (NST+HT+FB) for sparse signal recovery are exceedingly effective and fast, particularly for large scale problems. The core algorithm is shown to converge in finitely many steps under a (preconditioned) restricted isometry condition. In this paper, we present a new perspective to analyze the algorithm, which turns out that the efficiency of the algorithm can be further elaborated by an estimate of the number of iterations for the guaranteed convergence. The convergence condition of NST+HT+FB is also improved. Moreover, an adaptive scheme (AdptNST+HT+FB) without the knowledge of the sparsity level is proposed with its convergence guarantee. The number of iterations for the finite step of convergence of the AdptNST+HT+FB scheme is also derived. It is further shown that the number of iterations can be significantly reduced by exploiting the structure of the specific sparse signal or the random measurement matrix.

Convolutional Neural Networks (CNNs) and Vision Transformers (ViTs) stand as the two most popular foundation models for visual representation learning. While CNNs exhibit remarkable scalability with linear complexity w.r.t. image resolution, ViTs surpass them in fitting capabilities despite contending with quadratic complexity. A closer inspection reveals that ViTs achieve superior visual modeling performance through the incorporation of global receptive fields and dynamic weights. This observation motivates us to propose a novel architecture that inherits these components while enhancing computational efficiency. To this end, we draw inspiration from the recently introduced state space model and propose the Visual State Space Model (VMamba), which achieves linear complexity without sacrificing global receptive fields. To address the encountered direction-sensitive issue, we introduce the Cross-Scan Module (CSM) to traverse the spatial domain and convert any non-causal visual image into order patch sequences. Extensive experimental results substantiate that VMamba not only demonstrates promising capabilities across various visual perception tasks, but also exhibits more pronounced advantages over established benchmarks as the image resolution increases. Source code has been available at https://github.com/MzeroMiko/VMamba.

Vector Quantization (VQ) is a widely used method for converting continuous representations into discrete codes, which has become fundamental in unsupervised representation learning and latent generative models. However, VQ models are often hindered by the problem of representation collapse in the latent space, which leads to low codebook utilization and limits the scalability of the codebook for large-scale training. Existing methods designed to mitigate representation collapse typically reduce the dimensionality of latent space at the expense of model capacity, which do not fully resolve the core issue. In this study, we conduct a theoretical analysis of representation collapse in VQ models and identify its primary cause as the disjoint optimization of the codebook, where only a small subset of code vectors are updated through gradient descent. To address this issue, we propose SimVQ, a novel method which reparameterizes the code vectors through a linear transformation layer based on a learnable latent basis. This transformation optimizes the entire linear space spanned by the codebook, rather than merely updating the code vector selected by the nearest-neighbor search in vanilla VQ models. Although it is commonly understood that the multiplication of two linear matrices is equivalent to applying a single linear layer, our approach works surprisingly well in resolving the collapse issue in VQ models with just one linear layer. We validate the efficacy of SimVQ through extensive experiments across various modalities, including image and audio data with different model architectures. Our code is available at https://github.com/youngsheen/SimVQ.

Diffusion models suffer from slow sample generation at inference time. Therefore, developing a principled framework for fast deterministic/stochastic sampling for a broader class of diffusion models is a promising direction. We propose two complementary frameworks for accelerating sample generation in pre-trained models: Conjugate Integrators and Splitting Integrators. Conjugate integrators generalize DDIM, mapping the reverse diffusion dynamics to a more amenable space for sampling. In contrast, splitting-based integrators, commonly used in molecular dynamics, reduce the numerical simulation error by cleverly alternating between numerical updates involving the data and auxiliary variables. After extensively studying these methods empirically and theoretically, we present a hybrid method that leads to the best-reported performance for diffusion models in augmented spaces. Applied to Phase Space Langevin Diffusion [Pandey & Mandt, 2023] on CIFAR-10, our deterministic and stochastic samplers achieve FID scores of 2.11 and 2.36 in only 100 network function evaluations (NFE) as compared to 2.57 and 2.63 for the best-performing baselines, respectively. Our code and model checkpoints will be made publicly available at https://github.com/mandt-lab/PSLD.

To adapt a well-trained large model to downstream tasks, we propose constraining learning within its original latent space by leveraging linear combinations of its basis vectors. This approach ensures stable training without compromising the model's capabilities. Traditionally, constructing orthonormal bases from a matrix requires a transfer matrix, which significantly increases storage and computational overhead for parameters and feature maps. In this paper, we introduce Absorb and Decompose for Q, K, V, and O matrices, enabling their orthogonalization without the need for transfer matrices. Furthermore, the Absorb-Decompose operation eliminates redundant vectors, reducing the encoder attention parameters of Whisper-large-v3 by 46.42% without requiring additional training. For parameter-efficient and stable fine-tuning, we orthonormalized Q, K, V, and O and fine-tuned only the singular values, allowing efficient adaptation while constraining changes to the original latent space. When fine-tuning LLaMA-2-7B on eight commonsense reasoning datasets, our method outperforms LoRA by 5.4% and DoRA by 4.4%.

Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory and corresponding assumptions on the form of data. We propose Neural Fourier Transform (NFT), a general framework of learning the latent linear action of the group without assuming explicit knowledge of how the group acts on data. We present the theoretical foundations of NFT and show that the existence of a linear equivariant feature, which has been assumed ubiquitously in equivariance learning, is equivalent to the existence of a group invariant kernel on the dataspace. We also provide experimental results to demonstrate the application of NFT in typical scenarios with varying levels of knowledge about the acting group.

We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are assumed to be linear subspaces, this reduces to the classical problem of subspace clustering, which has been studied extensively over the past two decades. Unfortunately, many real-world datasets such as natural images can not be well approximated by linear subspaces. On the other hand, numerous works have attempted to learn an appropriate transformation of the data, such that data is mapped from a union of general non-linear manifolds to a union of linear subspaces (with points from the same manifold being mapped to the same subspace). However, many existing works have limitations such as assuming knowledge of the membership of samples to clusters, requiring high sampling density, or being shown theoretically to learn trivial representations. In this paper, we propose to optimize the Maximal Coding Rate Reduction metric with respect to both the data representation and a novel doubly stochastic cluster membership, inspired by state-of-the-art subspace clustering results. We give a parameterization of such a representation and membership, allowing efficient mini-batching and one-shot initialization. Experiments on CIFAR-10, -20, -100, and TinyImageNet-200 datasets show that the proposed method is much more accurate and scalable than state-of-the-art deep clustering methods, and further learns a latent linear representation of the data.

Latent-based image generative models, such as Latent Diffusion Models (LDMs) and Mask Image Models (MIMs), have achieved notable success in image generation tasks. These models typically leverage reconstructive autoencoders like VQGAN or VAE to encode pixels into a more compact latent space and learn the data distribution in the latent space instead of directly from pixels. However, this practice raises a pertinent question: Is it truly the optimal choice? In response, we begin with an intriguing observation: despite sharing the same latent space, autoregressive models significantly lag behind LDMs and MIMs in image generation. This finding contrasts sharply with the field of NLP, where the autoregressive model GPT has established a commanding presence. To address this discrepancy, we introduce a unified perspective on the relationship between latent space and generative models, emphasizing the stability of latent space in image generative modeling. Furthermore, we propose a simple but effective discrete image tokenizer to stabilize the latent space for image generative modeling. Experimental results show that image autoregressive modeling with our tokenizer (DiGIT) benefits both image understanding and image generation with the next token prediction principle, which is inherently straightforward for GPT models but challenging for other generative models. Remarkably, for the first time, a GPT-style autoregressive model for images outperforms LDMs, which also exhibits substantial improvement akin to GPT when scaling up model size. Our findings underscore the potential of an optimized latent space and the integration of discrete tokenization in advancing the capabilities of image generative models. The code is available at https://github.com/DAMO-NLP-SG/DiGIT.

Entropy and mutual information in neural networks provide rich information on the learning process, but they have proven difficult to compute reliably in high dimensions. Indeed, in noisy and high-dimensional data, traditional estimates in ambient dimensions approach a fixed entropy and are prohibitively hard to compute. To address these issues, we leverage data geometry to access the underlying manifold and reliably compute these information-theoretic measures. Specifically, we define diffusion spectral entropy (DSE) in neural representations of a dataset as well as diffusion spectral mutual information (DSMI) between different variables representing data. First, we show that they form noise-resistant measures of intrinsic dimensionality and relationship strength in high-dimensional simulated data that outperform classic Shannon entropy, nonparametric estimation, and mutual information neural estimation (MINE). We then study the evolution of representations in classification networks with supervised learning, self-supervision, or overfitting. We observe that (1) DSE of neural representations increases during training; (2) DSMI with the class label increases during generalizable learning but stays stagnant during overfitting; (3) DSMI with the input signal shows differing trends: on MNIST it increases, while on CIFAR-10 and STL-10 it decreases. Finally, we show that DSE can be used to guide better network initialization and that DSMI can be used to predict downstream classification accuracy across 962 models on ImageNet. The official implementation is available at https://github.com/ChenLiu-1996/DiffusionSpectralEntropy.

We tackle the problem disentangling the latent space of an autoencoder in order to separate labelled attribute information from other characteristic information. This then allows us to change selected attributes while preserving other information. Our method, matrix subspace projection, is much simpler than previous approaches to latent space factorisation, for example not requiring multiple discriminators or a careful weighting among their loss functions. Furthermore our new model can be applied to autoencoders as a plugin, and works across diverse domains such as images or text. We demonstrate the utility of our method for attribute manipulation in autoencoders trained across varied domains, using both human evaluation and automated methods. The quality of generation of our new model (e.g. reconstruction, conditional generation) is highly competitive to a number of strong baselines.

In deep learning, performance is strongly affected by the choice of architecture and hyperparameters. While there has been extensive work on automatic hyperparameter optimization for simple spaces, complex spaces such as the space of deep architectures remain largely unexplored. As a result, the choice of architecture is done manually by the human expert through a slow trial and error process guided mainly by intuition. In this paper we describe a framework for automatically designing and training deep models. We propose an extensible and modular language that allows the human expert to compactly represent complex search spaces over architectures and their hyperparameters. The resulting search spaces are tree-structured and therefore easy to traverse. Models can be automatically compiled to computational graphs once values for all hyperparameters have been chosen. We can leverage the structure of the search space to introduce different model search algorithms, such as random search, Monte Carlo tree search (MCTS), and sequential model-based optimization (SMBO). We present experiments comparing the different algorithms on CIFAR-10 and show that MCTS and SMBO outperform random search. In addition, these experiments show that our framework can be used effectively for model discovery, as it is possible to describe expressive search spaces and discover competitive models without much effort from the human expert. Code for our framework and experiments has been made publicly available.

We study the robust interpolation problem of arbitrary data distributions supported on a bounded space and propose a two-fold law of robustness. Robust interpolation refers to the problem of interpolating n noisy training data points in R^d by a Lipschitz function. Although this problem has been well understood when the samples are drawn from an isoperimetry distribution, much remains unknown concerning its performance under generic or even the worst-case distributions. We prove a Lipschitzness lower bound Omega(n/p) of the interpolating neural network with p parameters on arbitrary data distributions. With this result, we validate the law of robustness conjecture in prior work by Bubeck, Li, and Nagaraj on two-layer neural networks with polynomial weights. We then extend our result to arbitrary interpolating approximators and prove a Lipschitzness lower bound Omega(n^{1/d}) for robust interpolation. Our results demonstrate a two-fold law of robustness: i) we show the potential benefit of overparametrization for smooth data interpolation when n=poly(d), and ii) we disprove the potential existence of an O(1)-Lipschitz robust interpolating function when n=exp(omega(d)).

We present Manifold Diffusion Fields (MDF), an approach to learn generative models of continuous functions defined over Riemannian manifolds. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. Empirical results on several datasets and manifolds show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.

Temporal data such as time series can be viewed as discretized measurements of the underlying function. To build a generative model for such data we have to model the stochastic process that governs it. We propose a solution by defining the denoising diffusion model in the function space which also allows us to naturally handle irregularly-sampled observations. The forward process gradually adds noise to functions, preserving their continuity, while the learned reverse process removes the noise and returns functions as new samples. To this end, we define suitable noise sources and introduce novel denoising and score-matching models. We show how our method can be used for multivariate probabilistic forecasting and imputation, and how our model can be interpreted as a neural process.

Under low-light environment, handheld photography suffers from severe camera shake under long exposure settings. Although existing deblurring algorithms have shown promising performance on well-exposed blurry images, they still cannot cope with low-light snapshots. Sophisticated noise and saturation regions are two dominating challenges in practical low-light deblurring. In this work, we propose a novel non-blind deblurring method dubbed image and feature space Wiener deconvolution network (INFWIDE) to tackle these problems systematically. In terms of algorithm design, INFWIDE proposes a two-branch architecture, which explicitly removes noise and hallucinates saturated regions in the image space and suppresses ringing artifacts in the feature space, and integrates the two complementary outputs with a subtle multi-scale fusion network for high quality night photograph deblurring. For effective network training, we design a set of loss functions integrating a forward imaging model and backward reconstruction to form a close-loop regularization to secure good convergence of the deep neural network. Further, to optimize INFWIDE's applicability in real low-light conditions, a physical-process-based low-light noise model is employed to synthesize realistic noisy night photographs for model training. Taking advantage of the traditional Wiener deconvolution algorithm's physically driven characteristics and arisen deep neural network's representation ability, INFWIDE can recover fine details while suppressing the unpleasant artifacts during deblurring. Extensive experiments on synthetic data and real data demonstrate the superior performance of the proposed approach.