Roman Vershynin

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This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements – L1-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 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 -sparse signal in can be accurately estimated from m = O(s log(n/s)) single-bit measurements(More)
We study random submatrices of a large matrix <i>A</i>. We show how to approximately compute <i>A</i> from its random submatrix of the smallest possible size <i>O</i>(<i>r</i>log <i>r</i>) with a small error in the spectral norm, where <i>r</i> &equals; &Verbar;<i>A</i>&Verbar;<sup>2</sup><sub><i>F</i></sub>/&Verbar;<i>A</i>&Verbar;<sup>2</sup><sub>2</sub>(More)
We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n−1/2, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a(More)
Suppose we wish to transmit a vector f &#x003F5; R<sup>n</sup> 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 by an error e. We do not know which entries are affected nor do we know how they are affected. Is it(More)
We demonstrate a simple greedy algorithm that can reliably recover a vector v ∈ R 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 combines(More)