Message Passing Algorithms for Compressed Sensing: I. Motivation and Construction (0911.4219v1)

Abstract: In a paper, the authors proposed a new class of low-complexity iterative thresholding algorithms for reconstructing sparse signals from a small set of linear measurements \cite{DMM}. The new algorithms are broadly referred to as AMP, for approximate message passing. This is the first of two conference papers describing the derivation of these algorithms, connection with the related literature, extensions of the original framework, and new empirical evidence. In particular, the present paper outlines the derivation of AMP from standard sum-product belief propagation, and its extension in several directions. We also discuss relations with formal calculations based on statistical mechanics methods.

• The paper presents a novel derivation of AMP from belief propagation for efficient sparse signal recovery.
• It simplifies high-dimensional message updates to scalar parameters using Gaussian approximations to reduce computational overhead.
• The framework extends to Basis Pursuit Denoising (Lasso), enhancing applicability for large-scale, prior-informed signal reconstruction.

Overview of "Message Passing Algorithms for Compressed Sensing: I. Motivation and Construction"

This paper introduces a novel framework for compressed sensing by integrating message passing algorithms, particularly Approximate Message Passing (AMP). The authors aim to achieve effective signal reconstruction from limited linear measurements using AMP, an approach inspired by belief propagation techniques. The paper stands as the first of two discussing the derivation of AMP and its connections to existing methodologies, highlighting both theoretical underpinnings and practical extensions.

Mathematical Foundations and Derivation

The core problem addressed in the paper involves recovering a sparse signal $s_o$ from fewer observations than its dimensions, $y = As_o$, where $A$ is a measurement matrix. The traditional approach via $\ell_1$-minimization or basis pursuit ($\|s\|_1$ minimization) has limitations in large-scale applications due to the complexity of standard linear programming solvers. In contrast, iterative thresholding algorithms, though computationally efficient, historically showed inferior performance metrics compared to basis pursuit.

To bridge this gap, the authors derive AMP from the sum-product belief propagation under a probabilistic graphical model, presenting an alternative that retains the favorable computational complexity of iterative methods without sacrificing reconstruction accuracy. The derivation is methodically laid out, leveraging statistical mechanics to justify approximations that simplify message update rules in the large system limit.

Algorithmic Structure and Simplification

The AMP algorithm is efficiently condensed, necessitating updates of scalar parameters rather than high-dimensional message representations—significantly reducing computational overhead. The authors achieve further simplifications by approximating message distributions with Gaussian densities. By doing so, AMP capitalizes on both belief propagation and the efficiency of iterative thresholding.

Extension to Other Problems

In addition to addressing the basis pursuit problem, the paper extends the AMP framework to tackle the Basis Pursuit Denoising (BPDN) or Lasso problem, maintaining adaptability in scenarios with known signal distributions. This is particularly noteworthy as it broadens applicability and theoretical robustness, allowing integration of prior signal statistics when available. The resultant AMP algorithms are iteratively refined through threshold adjustments computed dynamically instead of predetermined parameters.

Related Work and Theoretical Implications

The integration of message passing algorithms into the field of compressed sensing is not unprecedented, yet previous attempts were hampered by computational impracticality and the challenge of defining accurate priors. This paper provides a comprehensive solution through its sum-product approach, complemented by insights from spin glass theory and statistical physics. Notably, the AMP algorithm aligns with the TAP equations in spin glasses, further validated by soft-independent state evolution.

The authors also position AMP within the broader narrative of statistical physics, aligning with the replica method's analytical predictions, thus cementing the theoretical soundness of AMP as an approximate inference technique. This cross-disciplinary linkage highlights potential advancements in algorithmic design, suggesting new directions for sparse recovery problems.

Implications and Speculations

AMP stands to significantly impact fields that rely on large-scale sparse recovery, such as medical imaging and seismic data interpretation. By marrying low complexity with high accuracy, it addresses a critical need across various disciplines. Future research may explore even broader classes of prior distributions, further enhancing recovery capabilities and robustness. Exploring AMP's applications in structured signal recovery and integration with emerging machine learning models could also open novel research avenues.

Overall, the presented work lays a solid foundation for subsequent developments and applications in compressed sensing and related signal processing tasks, inviting rigorous validation and real-world deployment.