A Simple Message-Passing Algorithm for Compressed Sensing

TL;DR

This paper introduces a simple message-passing algorithm for compressed sensing that efficiently recovers nonnegative vectors from binary measurement matrices, achieving near-optimal measurement counts and exact recovery for sparse signals.

Contribution

It presents a novel message-passing algorithm that works with bipartite graph-based measurement matrices, providing theoretical guarantees for approximate and exact recovery.

Findings

Achieves O(k log(n/k)) measurements for recovery

Recovers k-sparse vectors exactly in O(n log(n/k) log(k)) time

Provides approximation guarantees for non-sparse vectors

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.