text: This paper presents the recurrent estimation of distributions (RED) for modeling real-valued data in a semiparametric fashion. RED models make two novel uses of recurrent neural networks (RNNs) for density estimation of general real-valued data. First, RNNs are used to transform input covariates into a latent space to better capture conditional dependencies in inputs. After, an RNN is used to compute the conditional distributions of the latent covariates. The resulting model is efficient to train, compute, and sample from, whilst producing normalized pdfs. The effectiveness of RED is shown via several real-world data experiments. Our results show that RED models achieve a lower held-out negative log-likelihood than other neural network approaches across multiple dataset sizes and dimensionalities. Further context of the efficacy of RED is provided by considering anomaly detection tasks, where we also observe better performance over alternative models.
  @cite_14: 
  @cite_7: 
  @cite_8: The Neural Autoregressive Distribution Estimator (NADE) and its real-valued version RNADE are competitive density models of multidimensional data across a variety of domains. These models use a fixed, arbitrary ordering of the data dimensions. One can easily condition on variables at the beginning of the ordering, and marginalize out variables at the end of the ordering, however other inference tasks require approximate inference. In this work we introduce an efficient procedure to simultaneously train a NADE model for each possible ordering of the variables, by sharing parameters across all these models. We can thus use the most convenient model for each inference task at hand, and ensembles of such models with different orderings are immediately available. Moreover, unlike the original NADE, our training procedure scales to deep models. Empirically, ensembles of Deep NADE models obtain state of the art density estimation performance.
  @cite_1: Statistical parametric speech synthesis (SPSS) using deep neural networks (DNNs) has shown its potential to produce naturally-sounding synthesized speech. However, there are limitations in the current implementation of DNN-based acoustic modeling for speech synthesis, such as the unimodal nature of its objective function and its lack of ability to predict variances. To address these limitations, this paper investigates the use of a mixture density output layer. It can estimate full probability density functions over real-valued output features conditioned on the corresponding input features. Experimental results in objective and subjective evaluations show that the use of the mixture density output layer improves the prediction accuracy of acoustic features and the naturalness of the synthesized speech.
  @cite_6: We investigate the problem of modeling symbolic sequences of polyphonic music in a completely general piano-roll representation. We introduce a probabilistic model based on distribution estimators conditioned on a recurrent neural network that is able to discover temporal dependencies in high-dimensional sequences. Our approach outperforms many traditional models of polyphonic music on a variety of realistic datasets. We show how our musical language model can serve as a symbolic prior to improve the accuracy of polyphonic transcription.
  @cite_19: We propose a deep learning framework for modeling complex high-dimensional densities called Non-linear Independent Component Estimation (NICE). It is based on the idea that a good representation is one in which the data has a distribution that is easy to model. For this purpose, a non-linear deterministic transformation of the data is learned that maps it to a latent space so as to make the transformed data conform to a factorized distribution, i.e., resulting in independent latent variables. We parametrize this transformation so that computing the Jacobian determinant and inverse transform is trivial, yet we maintain the ability to learn complex non-linear transformations, via a composition of simple building blocks, each based on a deep neural network. The training criterion is simply the exact log-likelihood, which is tractable. Unbiased ancestral sampling is also easy. We show that this approach yields good generative models on four image datasets and can be used for inpainting.
  @cite_16: 
  @cite_10: Modeling the distribution of natural images is a landmark problem in unsupervised learning. This task requires an image model that is at once expressive, tractable and scalable. We present a deep neural network that sequentially predicts the pixels in an image along the two spatial dimensions. Our method models the discrete probability of the raw pixel values and encodes the complete set of dependencies in the image. Architectural novelties include fast two-dimensional recurrent layers and an effective use of residual connections in deep recurrent networks. We achieve log-likelihood scores on natural images that are considerably better than the previous state of the art. Our main results also provide benchmarks on the diverse ImageNet dataset. Samples generated from the model appear crisp, varied and globally coherent.
  @cite_20: 
summary: In order to provide greater flexibility with semiparametric models, recent work has employed deep learning for density estimation. The use of neural networks for density estimation dates back to early work by @cite_7 and has seen success in areas like speech @cite_1 @cite_16 , music @cite_6 , etc.. Typically such approaches use a network to learn to parameters of a parametric model for data. Recent work has also explored the application of deep learning to build density estimates in image data @cite_10 @cite_20 . However, such approaches are heavily reliant on exploiting structure in neighboring pixels, often reshaping or re-ordering data and using convolutions to take advantage of neighboring correlations. Modern approaches for general density estimation in real-valued data include neural autoregressive density estimation (NADE) @cite_8 @cite_14 and non-linear independent component estimation (NICE) @cite_19 . We briefly describe these approaches below and contrast them to RED models.