12 accepted papers
Learned denoisers play a fundamental role in various signal generation (e.g., diffusion models) and reconstruction (e.g., compressed sensing) architectures, whose success derives from their ability to leverage low-dimensional structure in data. Existing denoising methods, however, either rely on loc…
Data with low-dimensional nonlinear structure are ubiquitous in engineering and scientific problems. We study a model problem with such structure---a binary classification task that uses a deep fully-connected neural network to classify data drawn from two disjoint smooth curves on the unit sphere.…
We study the multiple manifold problem, a binary classification task modeled on applications in machine vision, in which a deep fully-connected neural network is trained to separate two low-dimensional submanifolds of the unit sphere. We provide an analysis of the one-dimensional case, proving for a…
We propose a new framework -- Square Root Principal Component Pursuit -- for low-rank matrix recovery from observations corrupted with noise and outliers. Inspired by the square root Lasso, this new formulation does not require prior knowledge of the noise level. We show that a single, universal cho…
Short-and-sparse deconvolution (SaSD) is the problem of extracting localized, recurring motifs in signals with spatial or temporal structure. Variants of this problem arise in applications such as image deblurring, microscopy, neural spike sorting, and more. The problem is challenging in both theory…
Randomly initialized first-order optimization algorithms are the method of choice for solving many high-dimensional nonconvex problems in machine learning, yet general theoretical guarantees cannot rule out convergence to critical points of poor objective value. For some highly structured nonconvex…
We study the Short-and-Sparse (SaS) deconvolution problem of recovering a short signal a0 and a sparse signal x0 from their convolution. We propose a method based on nonconvex optimization, which under certain conditions recovers the target short and sparse signals, up to a signed shift symmetry whi…
This paper considers compressed sensing (CS) in the context of RF spectrum sensing and presents an efficient approach for learning hardware nonidealities in an analog-to-information converter (A2IC). The proposed methodology is based on the learned iterative shrinkage-thresholding algorithm (LISTA),…
Blind deconvolution is a ubiquitous problem of recovering two unknown signals from their convolution. Unfortunately, this is an ill-posed problem in general. This paper focuses on the {\em short and sparse} blind deconvolution problem, where the one unknown signal is short and the other one is spars…
We study the convolutional phase retrieval problem, which asks us to recover an unknown signal ${\mathbf x} $ of length $n$ from $m$ measurements consisting of the magnitude of its cyclic convolution with a known kernel $\mathbf a$ of length $m$. This model is motivated by applications to channel es…
Blind deconvolution is the problem of recovering a convolutional kernel and an activation signal from their convolution. This problem is ill-posed without further constraints or priors. This paper studies the situation where the nonzero entries in the activation signal are sparsely and randomly popu…
We consider the problem of recovering a complete (i.e., square and invertible) dictionary mb A_0, from mb Y = mb A_0 mb X_0 with mb Y ∈\mathbb R^n \times p. This recovery setting is central to the theoretical understanding of dictionary learning. We give the first efficient algorithm that provably r…