Estimation with Random Linear Mixing, Belief Propagation and Compressed Sensing

TL;DR

This paper extends the relaxed belief propagation method to general measurement channels, enabling efficient estimation in compressed sensing and noisy environments with theoretical guarantees on its asymptotic behavior.

Contribution

It introduces a generalized relaxed BP algorithm for non-AWGN channels and proves its asymptotic equivalence to standard BP via state evolution equations.

Findings

Relaxed BP is computationally efficient for dense matrices.

The method applies to various channels beyond Gaussian noise.

Asymptotic analysis confirms identical behavior to standard BP.

Abstract

We apply Guo and Wang's relaxed belief propagation (BP) method to the estimation of a random vector from linear measurements followed by a componentwise probabilistic measurement channel. Relaxed BP uses a Gaussian approximation in standard BP to obtain significant computational savings for dense measurement matrices. The main contribution of this paper is to extend the relaxed BP method and analysis to general (non-AWGN) output channels. Specifically, we present detailed equations for implementing relaxed BP for general channels and show that relaxed BP has an identical asymptotic large sparse limit behavior as standard BP, as predicted by the Guo and Wang's state evolution (SE) equations. Applications are presented to compressed sensing and estimation with bounded noise.