Expectation-Maximization Gaussian-Mixture Approximate Message Passing

TL;DR

This paper introduces an empirical-Bayesian method that combines expectation-maximization with approximate message passing to learn the distribution of sparse signals modeled as Gaussian mixtures, improving recovery accuracy and efficiency.

Contribution

It proposes a novel approach that jointly learns the signal distribution and recovers the signal using AMP, enhancing performance over traditional methods like LASSO.

Findings

Achieves near-minimum MSE recovery in high-dimensional settings.

Outperforms LASSO in reconstruction error across various signal classes.

Offers computational efficiency and robustness for different sensing operators.

Abstract

When recovering a sparse signal from noisy compressive linear measurements, the distribution of the signal's non-zero coefficients can have a profound effect on recovery mean-squared error (MSE). If this distribution was apriori known, then one could use computationally efficient approximate message passing (AMP) techniques for nearly minimum MSE (MMSE) recovery. In practice, though, the distribution is unknown, motivating the use of robust algorithms like LASSO---which is nearly minimax optimal---at the cost of significantly larger MSE for non-least-favorable distributions. As an alternative, we propose an empirical-Bayesian technique that simultaneously learns the signal distribution while MMSE-recovering the signal---according to the learned distribution---using AMP. In particular, we model the non-zero distribution as a Gaussian mixture, and learn its parameters through expectation maximization, using AMP to implement the expectation step. Numerical experiments on a wide range of signal classes confirm the state-of-the-art performance of our approach, in both reconstruction error and runtime, in the high-dimensional regime, for most (but not all) sensing operators.