A Simple Message-Passing Algorithm for Compressed Sensing
Abstract
We consider the recovery of a nonnegative vector x from measurements y = Ax, where A is an m-by-n matrix whos entries are in {0, 1}. We establish that when A corresponds to the adjacency matrix of a bipartite graph with sufficient expansion, a simple message-passing algorithm produces an estimate \hat{x} of x satisfying ||x-\hat{x}||_1 \leq O(n/k) ||x-x(k)||_1, where x(k) is the best k-sparse approximation of x. The algorithm performs O(n (log(n/k))^2 log(k)) computation in total, and the number of measurements required is m = O(k log(n/k)). In the special case when x is k-sparse, the algorithm recovers x exactly in time O(n log(n/k) log(k)). Ultimately, this work is a further step in the direction of more formally developing the broader role of message-passing algorithms in solving compressed sensing problems.
Cite
@article{arxiv.1001.4110,
title = {A Simple Message-Passing Algorithm for Compressed Sensing},
author = {Venkat Chandar and Devavrat Shah and Gregory W. Wornell},
journal= {arXiv preprint arXiv:1001.4110},
year = {2010}
}