Statistical physics-based reconstruction in compressed sensing (1109.4424v4)

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.

• The paper demonstrates that a seeded belief propagation method, combined with innovative seeding matrices, pushes compressed sensing toward the information-theoretic optimal limit (M = K).
• It adapts message-passing algorithms and statistical physics techniques, such as replica analysis and density evolution, to overcome traditional signal recovery limitations.
• Numerical simulations validate that this method outperforms classic ℓ1-minimization, offering significant improvements for efficient data acquisition in practical applications.

Statistical-physics-based Reconstruction in Compressed Sensing

This paper discusses a novel approach in the compressed sensing domain, leveraging statistical physics to optimize signal reconstruction. The authors propose a technique that combines seeded Belief Propagation (s-BP), message-passing algorithms efficiently adapted from coding theory, and the innovative design of seeding measurement matrices. This synthesis allows the process to approach the theoretical limit, $M = K$, where $M$ is the number of measurements and $K$ is the number of non-zero components in a sparse signal, within large systems.

The mathematical framework revolves around belief propagation, a method previously shown effective in computational signal reconstruction in large, complex systems. The authors adapt it with seeding techniques to overcome existing limitations in compressed sensing, typically constrained by a requirement of excessively large measurement rates relative to signal density. Implementation is realized through specifically designed measurement matrices inspired by crystal nucleation theory. This novel matrix design effectively mitigates metastable states, a primary bottleneck hindering optimal signal reconstruction in previous works.

The seeded Belief Propagation (s-BP) approach described in the paper provides evidence of its potential to solve signal recovery problems at the 'information-theoretic optimal' rate. Numerical simulations conducted by the authors confirm its superiority over classic $\ell_1$-minimization techniques, such as those implemented in typical linear programming solutions. Their phase transition analysis, executed via statistical physics methodologies like the replica method and density evolution, highlights significant improvements in reconstruction thresholds. This represents a marked departure from prior approaches which required measurement rates greater than signal density for exact reconstructions.

In practical contexts, the findings imply several advancements for fields reliant on minimal-information data acquisition, such as MRI, systems biology, and single-pixel camera technologies. Given the structure and algorithmic scalability of s-BP, the method presents compelling potential for real-world applications where data acquisition is costly or infeasible at larger scales. The theoretical implications extend to an improved understanding of phase transitions in compressed sensing and the application of intricate statistical physics techniques to new domains of information processing.

Future developments are suggested regarding further optimization of the seeding matrices to accommodate various noise and signal distributions beyond those examined. Potential extensions also include adaptation for nonlinear measurements and incorporation of prior knowledge about signal distributions to enhance reconstruction efficiency.

This research signifies a substantial contribution to the compressed sensing field by challenging conventional limitations through interdisciplinary methodologies, revealing pathways to enhanced data acquisition and processing technologies.