text: Terahertz (THz) sensing is a promising imaging technology for a wide variety of different applications. Extracting the interpretable and physically meaningful parameters for such applications, however, requires solving an inverse problem in which a model function determined by these parameters needs to be fitted to the measured data. Since the underlying optimization problem is nonconvex and very costly to solve, we propose learning the prediction of suitable parameters from the measured data directly. More precisely, we develop a model-based autoencoder in which the encoder network predicts suitable parameters and the decoder is fixed to a physically meaningful model function, such that we can train the encoding network in an unsupervised way. We illustrate numerically that the resulting network is more than 140 times faster than classical optimization techniques while making predictions with only slightly higher objective values. Using such predictions as starting points of local optimization techniques allows us to converge to better local minima about twice as fast as optimization without the network-based initialization.
  @cite_29: This paper presents OptNet, a network architecture that integrates optimization problems (here, specifically in the form of quadratic programs) as individual layers in larger end-to-end train-able deep networks. These layers encode constraints and complex dependencies between the hidden states that traditional convolutional and fully-connected layers often cannot capture. In this paper, we explore the foundations for such an architecture: we show how techniques from sensitivity analysis, bilevel optimization, and implicit differentiation can be used to exactly differentiate through these layers and with respect to layer parameters; we develop a highly efficient solver for these layers that exploits fast GPU-based batch solves within a primal-dual interior point method, and which provides backpropagation gradients with virtually no additional cost on top of the solve; and we highlight the application of these approaches in several problems. In one notable example, we show that the method is capable of learning to play mini-Sudoku (4x4) given just input and output games, with no a priori information about the rules of the game; this highlights the ability of our architecture to learn hard constraints better than other neural architectures.
  @cite_19: 
  @cite_2: While variational methods have been among the most powerful tools for solving linear inverse problems in imaging, deep (convolutional) neural networks have recently taken the lead in many challenging benchmarks. A remaining drawback of deep learning approaches is their requirement for an expensive retraining whenever the specific problem, the noise level, noise type, or desired measure of fidelity changes. On the contrary, variational methods have a plug-and-play nature as they usually consist of separate data fidelity and regularization terms. In this paper we study the possibility of replacing the proximal operator of the regularization used in many convex energy minimization algorithms by a denoising neural network. The latter therefore serves as an implicit natural image prior, while the data term can still be chosen independently. Using a fixed denoising neural network in exemplary problems of image deconvolution with different blur kernels and image demosaicking, we obtain state-of-the-art reconstruction results. These indicate the high generalizability of our approach and a reduction of the need for problemspecific training. Additionally, we discuss novel results on the analysis of possible optimization algorithms to incorporate the network into, as well as the choices of algorithm parameters and their relation to the noise level the neural network is trained on.
  @cite_10: In this paper, we introduce variational networks (VNs) for image reconstruction. VNs are fully learned models based on the framework of incremental proximal gradient methods. They provide a natural transition between classical variational methods and state-of-the-art residual neural networks. Due to their incremental nature, VNs are very efficient, but only approximately minimize the underlying variational model. Surprisingly, in our numerical experiments on image reconstruction problems it turns out that giving up exact minimization leads to a consistent performance increase, in particular in the case of convex models.
  @cite_25: While deep learning methods have achieved state-of-theart performance in many challenging inverse problems like image inpainting and super-resolution, they invariably involve problem-specific training of the networks. Under this approach, each inverse problem requires its own dedicated network. In scenarios where we need to solve a wide variety of problems, e.g., on a mobile camera, it is inefficient and expensive to use these problem-specific networks. On the other hand, traditional methods using analytic signal priors can be used to solve any linear inverse problem; this often comes with a performance that is worse than learning-based methods. In this work, we provide a middle ground between the two kinds of methods — we propose a general framework to train a single deep neural network that solves arbitrary linear inverse problems. We achieve this by training a network that acts as a quasi-projection operator for the set of natural images and show that any linear inverse problem involving natural images can be solved using iterative methods. We empirically show that the proposed framework demonstrates superior performance over traditional methods using wavelet sparsity prior while achieving performance comparable to specially-trained networks on tasks including compressive sensing and pixel-wise inpainting.
  @cite_12: The move from hand-designed features to learned features in machine learning has been wildly successful. In spite of this, optimization algorithms are still designed by hand. In this paper we show how the design of an optimization algorithm can be cast as a learning problem, allowing the algorithm to learn to exploit structure in the problems of interest in an automatic way. Our learned algorithms, implemented by LSTMs, outperform generic, hand-designed competitors on the tasks for which they are trained, and also generalize well to new tasks with similar structure. We demonstrate this on a number of tasks, including simple convex problems, training neural networks, and styling images with neural art.
summary: Due to the revolutionary success (convolutional) neural networks have had on computer vision problems over the last decade, researchers have extended the fields of applications of neural networks significantly. A particularly interesting concept is to learn the solution of complex, possibly nonconvex, optimization problems. Different lines of research have considered directly learning the optimizer itself, e.g. modelled as a recurrent neural network @cite_12 , or rolling out optimization algorithms and learning the incremental steps, e.g. in the form of parameterized proximal operators in @cite_10 . Further hybrid approaches include optimization problems in the networks' architecture, e.g. @cite_29 , or combining optimizers with networks that have been trained individually @cite_25 @cite_2 . The recent work of @cite_19 trains a network to predict descent directions to a given energy in order to give provable convergence results on the learned optimizer.