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We present a statistical framework to benchmark the performance of reconstruction algorithms for linear inverse problems, in particular, neural-network-based methods that require large quantities of training data. We generate synthetic…
We consider the least-squares regression problem and provide a detailed asymptotic analysis of the performance of averaged constant-step-size stochastic gradient descent (a.k.a. least-mean-squares). In the strongly-convex case, we provide…
We obtain minimax-optimal convergence rates in the supremum norm, including information-theoretic lower bounds, for estimating the covariance kernel of a stochastic process which is repeatedly observed at discrete, synchronous design…
Scientific research and engineering practice often require the modeling and decomposition of nonlinear systems. The Dynamic Mode Decomposition (DMD) is a novel Koopman-based technique that effectively dissects high-dimensional nonlinear…
This paper is concerned with the problem of reconstructing an infinite-dimensional signal from a limited number of linear measurements. In particular, we show that for binary measurements (modelled with Walsh functions and Hadamard…
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,…
The Sobolev space $H^{s}(\mathbb{R}^{d})$, where $s > d/2$, is an important function space that has many applications in various areas of research. Attributed to the inertia of a measuring instrument, it is desirable in sampling theory to…
The sparse polynomial approximation of continuous functions has emerged as a prominent area of interest in function approximation theory in recent years. A key challenge within this domain is the accurate estimation of approximation errors.…
Generative compressed sensing uses the range of a pretrained generator as a nonlinear model for recovering structured signals from limited measurements. We study a conditional version of this problem for image recovery from subsampled…
Deep neural networks have had an enormous impact on image analysis. State-of-the-art training methods, based on weight decay and DropOut, result in impressive performance when a very large training set is available. However, they tend to…
Sampled-data (SD) systems, which are composed of both discrete- and continuous-time components, are arguably one of the most common classes of cyberphysical systems in practice; most modern controllers are implemented on digital platforms…
Kernels are efficient in representing nonlocal dependence and they are widely used to design operators between function spaces. Thus, learning kernels in operators from data is an inverse problem of general interest. Due to the nonlocal…
Training deep neural networks typically relies on backpropagating high dimensional error signals a computationally intensive process with little evidence supporting its implementation in the brain. However, since most tasks involve…
Least-squares approximation is one of the most important methods for recovering an unknown function from data. While in many applications the data is fixed, in many others there is substantial freedom to choose where to sample. In this…
We propose a reconstruction-based a posteriori error estimate for linear advection problems in one space dimension. In our framework, a stable variational ultra-weak formulation is adopted, and the equivalence of the $L_2$-norm of the error…
We develop a sampling scheme on the sphere that permits accurate computation of the spherical harmonic transform and its inverse for signals band-limited at $L$ using only $L^2$ samples. We obtain the optimal number of samples given by the…
We investigate the stability of vector recovery from random linear measurements which have been either clipped or folded. This is motivated by applications where measurement devices detect inputs outside of their effective range. As…
Sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, the exact sampling…
Recently, several works have shown that natural modifications of the classical conditional gradient method (aka Frank-Wolfe algorithm) for constrained convex optimization, provably converge with a linear rate when: i) the feasible set is a…
Algorithmic stability is a classical framework for analyzing the generalization error of learning algorithms. It predicts that an algorithm has small generalization error if it is insensitive to small perturbations in the training set such…