EE 381 V : Large Scale Learning Spring 2013 Lecture 21 — April 2

In the past few lectures we have been studying the problem of sparse signal reconstruction from a small number of noiseless linear measurements. The objective is to recover an unknown, k-sparse, n-dimensional vector x from a measurement vector y = Ax ∈ R, which is a linear transformation of x by a known m× n matrix A, where m < n. We have presented various heuristics for sparse signal recovery, including greedy algorithms such as Orthogonal Matching Pursuit, and algorithms based on convex optimization such as Basis Pursuit. In order to provide performance guarantees, the analysis of these algorithms seeks conditions on the measurement matrix A and the sparsity k of x, under which the solution is unique and/or the algorithm can successfully recover the sparsest solution. Examples of such conditions include mutual coherence (maximum coherence between columns of the measurement matrix A), the Restricted Isometry Property, and the Restricted Strong Convexity (which may be the subject of a future lecture). The previous lecture focused on the analysis of Basis Pursuit under the assumption that the measurement matrix satisfies an RIP condition. Specifically, if A satisfies the ( , 2k)RIP for < √ 2− 1, then we can recover the unique k-sparse vector x as the solution of the l1-minimization problem