Machine learning techniques have been successfully applied to super-resolution tasks on natural images where visually pleasing results are sufficient. However in many scientific domains this is not adequate and estimations of errors and uncertainties are crucial. To address this issue we propose a Bayesian framework that decomposes uncertainties into epistemic and aleatoric uncertainties. We test the validity of our approach by super-resolving images of the Sun's magnetic field and by generating maps measuring the range of possible high resolution explanations compatible with a given low resolution magnetogram.

This paper considers the problem of Phase Identification in power distribution systems. In particular, it focuses on improving supervised learning accuracies by focusing on exploiting some of the problem's information theoretic properties. This focus, along with recent advances in Information Theoretic Machine Learning (ITML), helps us to create two new techniques. The first transforms a bound on information losses into a data selection technique. This is important because phase identification data labels are difficult to obtain in practice. The second interprets the properties of distribution systems in the terms of ITML. This allows us to obtain an improvement in the representation learned by any classifier applied to the problem. We tested these two techniques experimentally on real datasets and have found that they yield phenomenal performance in every case. In the most extreme case, they improve phase identification accuracy from $51.7\%$ to $97.3\%$. Furthermore, since many problems share the physical properties of phase identification exploited in this paper, the techniques can be applied to a wide range of similar problems.

We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution $\nu = e^{-f}$ on $\mathbb{R}^n$ under isoperimetry. We prove a convergence guarantee for PLA in Kullback-Leibler (KL) divergence when $\nu$ satisfies log-Sobolev inequality (LSI) and $f$ has bounded second and third derivatives. This improves on the result for the Unadjusted Langevin Algorithm (ULA), and matches the fastest known rate for sampling under LSI (without Metropolis filter) with a better dependence on the LSI constant. We also prove convergence guarantees for PLA in R\'enyi divergence of order $q > 1$ when the biased limit satisfies either LSI or Poincar\'e inequality.

We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let $\mathsf{opt} < 1$ be the population loss of the best-fitting ReLU. We prove: 1. Finding a ReLU with square-loss $\mathsf{opt} + \epsilon$ is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply -{\emph unconditionally}- that gradient descent cannot converge to the global minimum in polynomial time. 2. There exists an efficient approximation algorithm for finding the best-fitting ReLU that achieves error $O(\mathsf{opt}^{2/3})$. The algorithm uses a novel reduction to noisy halfspace learning with respect to $0/1$ loss. Prior work due to Soltanolkotabi [Sol17] showed that gradient descent can find the best-fitting ReLU with respect to Gaussian marginals, if the training set is exactly labeled by a ReLU.

ARXIV 1 Dual-domain Cascade of U-nets for Multi-channel Magnetic Resonance Image Reconstruction Roberto Souza, PhD, Mariana Bento, PhD, Nikita Nogovitsyn, MSc, MD, Kevin J. Chung, BSc, R. Marc Lebel, PhD, and Richard Frayne, PhD Abstract --The U-net is a deep-learning network model that has been used to solve a number of inverse problems. In this work, the concatenation of two-element U-nets, termed the W-net, operating in k-space (K) and image (I) domains, were evaluated for multi-channel magnetic resonance (MR) image reconstruction. The two element network combinations were evaluated for the four possible image-k-space domain configurations: a) W-net II, b) W-net KK, c) W-net IK, and d) W-net KI were evaluated. Selected promising four element networks (WW-nets) were also examined. Two configurations of each network were compared: 1) Each coil channel processed independently, and 2) all channels processed simultaneously. One hundred and eleven volumetric, T1-weighted, 12-channel coil k-space datasets were used in the experiments. Normalized root mean squared error, peak signal to noise ratio, visual information fidelity and visual inspection were used to assess the reconstructed images against the fully sampled reference images. Our results indicated that networks that operate solely in the image domain are better suited when processing individual channels of multi-channel data independently. Dual domain methods are more advantageous when simultaneously reconstructing all channels of multi-channel data. Also, the appropriate cascade of U-nets compared favorably ( p 0 . Index T erms --Magnetic resonance imaging, compressed sensing, multi-channel (coil), image reconstruction, inverse problems, brain, machine learning M AGNETIC RESONANCE (MR) imaging is a sensitive diagnostic modality that allows specific, high-quality investigation of structure and function of the brain and body. One major drawback is the overall acquisition time to complete an MR imaging protocol, which can easily exceed 30 minutes per patient [1]. Lengthy MR examination times are costly ( 300 USD or more per examination); increase susceptibility to patient motion artifacts, which negatively impact image quality; further reduce patient throughput and contribute to repeated studies.

We propose a novel training procedure for improving the performance of generative adversarial networks (GANs), especially to bidirectional GANs. First, we enforce that the empirical distribution of the inverse inference network matches the prior distribution, which favors the generator network reproducibility on the seen samples. Second, we have found that the marginal log-likelihood of the samples shows a severe overrepresentation of a certain type of samples. To address this issue, we propose to train the bidirectional GAN using a non-uniform sampling for the mini-batch selection, resulting in improved quality and variety in generated samples measured quantitatively and by visual inspection. We illustrate our new procedure with the well-known CIFAR10, Fashion MNIST and CelebA datasets.

Does over-parameterization eliminate sub-optimal local minima for neural network problems? On one hand, existing positive results do not prove the claim, but often weaker claims. On the other hand, existing negative results have strong assumptions on the activation functions and/or data samples, causing a large gap with positive results. It was unclear before whether there is a clean answer of "yes" or "no". In this paper, we answer this question with a strong negative result. In particular, we prove that for deep and over-parameterized networks, sub-optimal local minima exist for generic input data samples and generic nonlinear activation. This is the setting widely studied in the global landscape of over-parameterized networks, thus our result corrects a possible misconception that "over-parameterization eliminates sub-optimal local-min". Our construction is based on fundamental optimization analysis, and thus rather principled.

We develop amortized population Gibbs (APG) samplers, a new class of autoencoding variational methods for deep probabilistic models. APG samplers construct high-dimensional proposals by iterating over updates to lower-dimensional blocks of variables. Each conditional update is a neural proposal, which we train by minimizing the inclusive KL divergence relative to the conditional posterior. To appropriately account for the size of the input data, we develop a new parameterization in terms of neural sufficient statistics, resulting in quasi-conjugate variational approximations. Experiments demonstrate that learned proposals converge to the known analytical conditional posterior in conjugate models, and that APG samplers can learn inference networks for highly-structured deep generative models when the conditional posteriors are intractable. Here APG samplers offer a path toward scaling up stochastic variational methods to models in which standard autoencoding architectures fail to produce accurate samples.

We introduce a gradient-based learning method to automatically adapt Markov chain Monte Carlo (MCMC) proposal distributions to intractable targets. We define a maximum entropy regularised objective function, referred to as generalised speed measure, which can be robustly optimised over the parameters of the proposal distribution by applying stochastic gradient optimisation. An advantage of our method compared to traditional adaptive MCMC methods is that the adaptation occurs even when candidate state values are rejected. This is a highly desirable property of any adaptation strategy because the adaptation starts in early iterations even if the initial proposal distribution is far from optimum. We apply the framework for learning multivariate random walk Metropolis and Metropolis-adjusted Langevin proposals with full covariance matrices, and provide empirical evidence that our method can outperform other MCMC algorithms, including Hamiltonian Monte Carlo schemes.

Consider a loss function G ( a,θ) ( a,θ) G ( a,θ) R, where a A R s is a decision/design variable and θ Θ R d is a model parameter space. Given a set of observations X n {ξ 1,...,ξ n} drawn from a distribution with unknown parameter θ 0, p( X n θ 0), our goal is to compute the Bayes optimal decision rule a ( X n) arg min a A E π[G ( a,θ)] ΘG ( a,θ) π ( θ X n) dθ, (1) where π ( θ X n) is the posterior distribution. The latter results when a Bayesian decision-maker places a prior distribution π ( θ) over the parameter space Θ, capturing a priori information about θ such as location or spread. Given X n, the prior and likelihood p ( X n θ) together define a posterior distribution π ( θ X n) p ( X n θ) π ( θ) p( θ, X n), the conditional distribution over θ given observations. The posterior distribution represents uncertainty over the unknown parameter θ, and contains all information required for further inferences or optimization. In general, under most realistic modeling assumptions, closed-form analytic expressions are unavailable for π ( θ X n), making the subsequent integration and optimization problems intractable. In practice, therefore, one uses an approximation to the posterior in the integration in (1). It is easy to see that posterior computation can be expressed as a convex optimization problem: min q () M KL( q ( θ) π ( θ X n)) KL( q ( θ) p ( θ, X n)) log p( X n) (2) KL( q ( θ) π ( θ)) Θlog p( X n θ) q ( θ) dθ log p ( X n) where KL is the Kullback-Leibler divergence and M is the space of all distributions that are absolutely continuous with respect to the posterior (or, equivalently, the prior).