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2 3 This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of these methods sprung off from the development of geometric functional analysis since the 1970's. They have… (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 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)

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 n from the signs of O(s log 2 (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 study the problem of signal estimation from non-linear observations when the signal belongs to a low-dimensional set buried in a high-dimensional space. A rough heuristic often used in practice postulates that non-linear observations may be treated as noisy linear observations, and thus the signal may be estimated using the generalized Lasso. This is… (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> = ‖<i>A</i>‖<sup>2</sup><sub><i>F</i></sub>/‖<i>A</i>‖<sup>2</sup><sub>2</sub>… (More)

Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y ∈ K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they… (More)

We present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs. The method is based on Grothendieck's inequality. Unlike the previous uses of this inequality that lead to constant relative accuracy, we achieve any given relative accuracy by leveraging randomness. We illustrate the method with the problem… (More)