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Several recent works have empirically observed that Convolutional Neural Nets (CNNs) are (approximately) invertible. To understand this approximate invertibility phenomenon and how to leverage it more effectively, we focus on a theoretical…

Regularization plays an important role in solving ill-posed problems by adding extra information about the desired solution, such as sparsity. Many regularization terms usually involve some vector norm, e.g., $L_1$ and $L_2$ norms. In this…

The problem of estimating a sparse signal from low dimensional noisy observations arises in many applications, including super resolution, signal deconvolution, and radar imaging. In this paper, we consider a sparse signal model with…

In this survey we describe some modifications of Prony's method. In particular, we consider the recovery of general expansions into eigenfunctions of linear differential operators of first order and show, how these expansions can be…

We derive non-asymptotic bounds for the minimax risk of variable selection under expected Hamming loss in the Gaussian mean model in $\mathbb{R}^d$ for classes of $s$-sparse vectors separated from 0 by a constant $a > 0$. In some cases, we…

We establish a general form of explicit, input-dependent, measure-valued warpings for learning nonstationary kernels. While stationary kernels are ubiquitous and simple to use, they struggle to adapt to functions that vary in smoothness…

Dictionary learning can be used for image superresolution by learning a pair of coupled dictionaries of image patches from high-resolution and low-resolution image pairs such that the corresponding pairs share the same sparse vector when…

The purpose of this paper is to propose a non-iterative method for the inverse conductivity problem of recovering multiple small anomalies from the boundary measurements. When small anomalies are buried in a conducting object, the electric…

In computational optical imaging and wireless communications, signals are acquired through linear coded and noisy projections, which are recovered through computational algorithms. Deep model-based approaches, i.e., neural networks…

Sparse linear regression is a central problem in high-dimensional statistics. We study the correlated random design setting, where the covariates are drawn from a multivariate Gaussian $N(0,\Sigma)$, and we seek an estimator with small…

The convergence theory for the gradient sampling algorithm is extended to directionally Lipschitz functions. Although directionally Lipschitz functions are not necessarily locally Lipschitz, they are almost everywhere differentiable and…

Uncertainty sampling is a prevalent active learning algorithm that queries sequentially the annotations of data samples which the current prediction model is uncertain about. However, the usage of uncertainty sampling has been largely…

We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence…

The success of the compressed sensing paradigm has shown that a substantial reduction in sampling and storage complexity can be achieved in certain linear and non-adaptive estimation problems. It is therefore an advisable strategy for…

Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear…

Audio events are quite often overlapping in nature, and more prone to noise than visual signals. There has been increasing evidence for the superior performance of representations learned using sparse dictionaries for applications like…

Given an overcomplete dictionary $A$ and a signal $b$ that is a linear combination of a few linearly independent columns of $A$, classical sparse recovery theory deals with the problem of recovering the unique sparse representation $x$ such…

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents…

Soft extrapolation refers to the problem of recovering a function from its samples, multiplied by a fast-decaying window and perturbed by an additive noise, over an interval which is potentially larger than the essential support of the…

In this paper, we analyse the recovery properties of nonconvex regularized $M$-estimators, under the assumption that the true parameter is of soft sparsity. In the statistical aspect, we establish the recovery bound for any stationary point…