Learn More
This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements – L 1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and(More)
This paper improves upon best-known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly nonconvex problem to a convex problem and then solve it as a linear program. We show(More)
The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We introduce a randomized version of the Kaczmarz method for(More)
This paper develops theoretical results regarding noisy 1-bit compressed sensing and sparse binomial regression. We demonstrate that a single convex program gives an accurate estimate of the signal, or coefficient vector, for both of these models. We show that an s-sparse signal in R n can be accurately estimated from m = O(s log(n/s)) single-bit(More)
— Given a frame in C n which satisfies a form of the uncertainty principle (as introduced by Candes and Tao), it is shown how to quickly convert the frame representation of every vector into a more robust Kashin's representation whose coefficients all have the smallest possible dynamic range O(1/ √ n). The information tends to spread evenly among these(More)
Suppose we wish to transmit a vector f ∈ R n reliably. A frequently discussed approach consists in encoding f with an m by n coding matrix A. Assume now that a fraction of the entries of Af are corrupted in a completely arbitrary fashion. We do not know which entries are affected nor do we know how they are affected. Is it possible to recover f exactly from(More)
The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to information theory operate with random matrices in fixed dimensions. This survey addresses the non-asymptotic theory of(More)
We demonstrate a simple greedy algorithm that can reliably recover a vector v ∈ R d from incomplete and inaccurate measurements x = Φv + e. Here Φ is a N × d measurement matrix with N ≪ d, and e is an error vector. Our algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to close the gap between two major approaches to sparse recovery. It(More)