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- Roman Vershynin
- ArXiv
- 2010

2 Preliminaries 7 2.1 Matrices and their singular values . . . . . . . . . . . . . . . . . . 7 2.2 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Sub-gaussian random variables . . . . . . . . . . . . . . . . . . . 9 2.4 Sub-exponential random variables . . . . . . . . . . . . . . . . . . 14 2.5 Isotropic random vectors . . .… (More)

- Deanna Needell, Roman Vershynin
- Foundations of Computational Mathematics
- 2009

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 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)

- Yaniv Plan, Roman Vershynin
- IEEE Transactions on Information Theory
- 2013

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)

- Mark Rudelson, Roman Vershynin
- J. ACM
- 2007

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> = ‖<i>A</i>‖<sup>2</sup><sub><i>F</i></sub>/‖<i>A</i>‖<sup>2</sup><sub>2</sub>… (More)

- Yaniv Plan, Roman Vershynin
- ArXiv
- 2011

We give the first computationally tractable and almost optimal solution to the problem of one-bit compressed sensing, showing how to accurately recover an s-sparse vector x ∈ R from the signs of O(s log(n/s)) random linear measurements of x. The recovery is achieved by a simple linear program. This result extends to approximately sparse vectors x. Our… (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)

- Emmanuel J. Candès, Mark Rudelson, Terence Tao, Roman Vershynin
- 46th Annual IEEE Symposium on Foundations of…
- 2005

Suppose we wish to transmit a vector f ϵ 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)