Statistical physics-based reconstruction in compressed sensing

TL;DR

This paper introduces a novel compressed sensing reconstruction method leveraging statistical physics, belief propagation, and optimized measurement matrices, achieving near-theoretical measurement efficiency for large systems.

Contribution

It presents a new reconstruction algorithm that approaches the theoretical measurement limit using a probabilistic, message-passing approach with physics-inspired measurement design.

Findings

Exact reconstruction near the theoretical measurement limit

Algorithm outperforms existing methods in large systems

Numerical validation confirms theoretical predictions

Abstract

Compressed sensing is triggering a major evolution in signal acquisition. It consists in sampling a sparse signal at low rate and later using computational power for its exact reconstruction, so that only the necessary information is measured. Currently used reconstruction techniques are, however, limited to acquisition rates larger than the true density of the signal. We design a new procedure which is able to reconstruct exactly the signal with a number of measurements that approaches the theoretical limit in the limit of large systems. It is based on the joint use of three essential ingredients: a probabilistic approach to signal reconstruction, a message-passing algorithm adapted from belief propagation, and a careful design of the measurement matrix inspired from the theory of crystal nucleation. The performance of this new algorithm is analyzed by statistical physics methods. The obtained improvement is confirmed by numerical studies of several cases.