We introduce a novel deep-learning inspired formulation of the \textit{phase retrieval problem}, which asks to recover a signal $y_0 \in \R^n$ from $m$ quadratic observations, under structural assumptions on the underlying signal. As is common in many imaging problems, previous methodologies have considered natural signals as being sparse with respect to a known basis, resulting in the decision to enforce a generic sparsity prior. However, these methods for phase retrieval have encountered possibly fundamental limitations, as no computationally efficient algorithm for sparse phase retrieval has been proven to succeed with fewer than $O(k^2\log n)$ generic measurements, which is larger than the theoretical optimum of $O(k \log n)$. In this paper, we sidestep this issue by considering a prior that a natural signal is in the range of a generative neural network $G: \R^k \rightarrow \R^n$. We introduce an empirical risk formulation that has favorable global geometry for gradient methods, as soon as $m = O(k)$, under the model of a multilayer fully-connected neural network with random weights. Specifically, we show that there exists a descent direction outside of a small neighborhood around the true $k$-dimensional latent code and a negative multiple thereof. This formulation for structured phase retrieval thus benefits from two effects: generative priors can more tightly represent natural signals than sparsity priors, and this empirical risk formulation can exploit those generative priors at an information theoretically optimal sample complexity, unlike for a sparsity prior. We corroborate these results with experiments showing that exploiting generative models in phase retrieval tasks outperforms both sparse and general phase retrieval methods.

Meanwhile, we are still waiting on that long-anticipated sequel. A judge has ruled that publisher Krafton must reinstate Ted Gill as CEO of Unknown Worlds Entertainment, . The company fired Gill and two other co-founders last year as part of a shakeup . The Delaware judge said Krafton had violated the terms of its contract with Unknown Worlds when it fired the executives. To remedy these breaches, Gill is reinstated as CEO of Unknown Worlds with full operational authority over the studio, wrote judge Lori W. Will.

We present a novel model architecture which leverages deep learning tools to perform exact Bayesian inference on sets of high dimensional, complex observations. Our model is provably exchangeable, meaning that the joint distribution over observations is invariant under permutation: this property lies at the heart of Bayesian inference. The model does not require variational approximations to train, and new samples can be generated conditional on previous samples, with cost linear in the size of the conditioning set. The advantages of our architecture are demonstrated on learning tasks that require generalisation from short observed sequences while modelling sequence variability, such as conditional image generation, few-shot learning, and anomaly detection.

Recent work has shown that fast, compact low-bitwidth neural networks can be surprisingly accurate. These networks use homogeneous binarization: all parameters in each layer or (more commonly) the whole model have the same low bitwidth (e.g., 2 bits). However, modern hardware allows efficient designs where each arithmetic instruction can have a custom bitwidth, motivating heterogeneous binarization, where every parameter in the network may have a different bitwidth. In this paper, we show that it is feasible and useful to select bitwidths at the parameter granularity during training. For instance a heterogeneously quantized version of modern networks such as AlexNet and MobileNet, with the right mix of 1-, 2-and 3-bit parameters that average to just 1.4 bits can equal the accuracy of homogeneous 2-bit versions of these networks. Further, we provide analyses to show that the heterogeneously binarized systems yield FPGA-and ASIC-based implementations that are correspondingly more efficient in both circuit area and energy efficiency than their homogeneous counterparts.

In this paper, we study the problems of principle Generalized Eigenvector computation and Canonical Correlation Analysis in the stochastic setting. We propose a simple and efficient algorithm for these problems. We prove the global convergence of our algorithm, borrowing ideas from the theory of fast-mixing Markov chains and two-Time-Scale Stochastic Approximation, showing that it achieves the optimal rate of convergence. In the process, we develop tools for understanding stochastic processes with Markovian noise which might be of independent interest.

Time series classification using deep neural networks, such as convolutional neural networks (CNN), operate on the spectral decomposition of the time series computed using a preprocessing step. This step can include a large number of hyperparameters, such as window length, filter widths, and filter shapes, each with a range of possible values that must be chosen using time and data intensive cross-validation procedures. We propose the wavelet deconvolution (WD) layer as an efficient alternative to this preprocessing step that eliminates a significant number of hyperparameters. The WD layer uses wavelet functions with adjustable scale parameters to learn the spectral decomposition directly from the signal. Using backpropagation, we show the scale parameters can be optimized with gradient descent. Furthermore, the WD layer adds interpretability to the learned time series classifier by exploiting the properties of the wavelet transform.

Optimizing task-related mathematical model is one of the most fundamental methodologies in statistic and learning areas. However, generally designed schematic iterations may hard to investigate complex data distributions in real-world applications. Recently, training deep propagations (i.e., networks) has gained promising performance in some particular tasks. Unfortunately, existing networks are often built in heuristic manners, thus lack of principled interpretations and solid theoretical supports. In this work, we provide a new paradigm, named Propagation and Optimization based Deep Model (PODM), to bridge the gaps between these different mechanisms (i.e., model optimization and deep propagation). On the one hand, we utilize PODM as a deeply trained solver for model optimization. Different from these existing network based iterations, which often lack theoretical investigations, we provide strict convergence analysis for PODM in the challenging nonconvex and nonsmooth scenarios. On the other hand, by relaxing the model constraints and performing end-to-end training, we also develop a PODM based strategy to integrate domain knowledge (formulated as models) and real data distributions (learned by networks), resulting in a generic ensemble framework for challenging real-world applications. Extensive experiments verify our theoretical results and demonstrate the superiority of PODM against these state-of-the-art approaches.

The performance of neural networks on high-dimensional data distributions suggests that it may be possible to parameterize a representation of a given high-dimensional function with controllably small errors, potentially outperforming standard interpolation methods. We demonstrate, both theoretically and numerically, that this is indeed the case. We map the parameters of a neural network to a system of particles relaxing with an interaction potential determined by the loss function. We show that in the limit that the number of parameters $n$ is large, the landscape of the mean-squared error becomes convex and the representation error in the function scales as $O(n^{-1})$. In this limit, we prove a dynamical variant of the universal approximation theorem showing that the optimal representation can be attained by stochastic gradient descent, the algorithm ubiquitously used for parameter optimization in machine learning. In the asymptotic regime, we study the fluctuations around the optimal representation and show that they arise at a scale $O(n^{-1})$. These fluctuations in the landscape identify the natural scale for the noise in stochastic gradient descent. Our results apply to both single and multi-layer neural networks, as well as standard kernel methods like radial basis functions.

An important factor contributing to the success of deep learning has been the remarkable ability to optimize large neural networks using simple first-order optimization algorithms like stochastic gradient descent. While the efficiency of such methods depends crucially on the local curvature of the loss surface, very little is actually known about how this geometry depends on network architecture and hyperparameters. In this work, we extend a recently-developed framework for studying spectra of nonlinear random matrices to characterize an important measure of curvature, namely the eigenvalues of the Fisher information matrix. We focus on a single-hidden-layer neural network with Gaussian data and weights and provide an exact expression for the spectrum in the limit of infinite width. We find that linear networks suffer worse conditioning than nonlinear networks and that nonlinear networks are generically non-degenerate. We also predict and demonstrate empirically that by adjusting the nonlinearity, the spectrum can be tuned so as to improve the efficiency of first-order optimization methods.

We investigate the efficiency of k-means in terms of both statistical and computational requirements. More precisely, we study a Nystr\om approach to kernel k-means. We analyze the statistical properties of the proposed method and show that it achieves the same accuracy of exact kernel k-means with only a fraction of computations. Indeed, we prove under basic assumptions that sampling $\sqrt{n}$ Nystr\om landmarks allows to greatly reduce computational costs without incurring in any loss of accuracy. To the best of our knowledge this is the first result showing in this kind for unsupervised learning.