Improved Turbo Message Passing for Compressive Robust Principal Component Analysis: Algorithm Design and Asymptotic Analysis

TL;DR

This paper introduces an improved turbo message passing algorithm for compressive robust PCA, providing a probabilistic model, asymptotic analysis, and convergence conditions validated by numerical experiments.

Contribution

The paper develops an enhanced ITMP algorithm for CRPCA with a new probabilistic model and asymptotic analysis framework, including convergence conditions.

Findings

The asymptotic analysis accurately predicts the algorithm's behavior.

The phase transition curve matches numerical simulations.

The convergence conditions are sufficient for global convergence.

Abstract

Compressive Robust Principal Component Analysis (CRPCA) naturally arises in various applications as a means to recover a low-rank matrix low-rank matrix $\boldsymbol{L}$ and a sparse matrix $\boldsymbol{S}$ from compressive measurements. In this paper, we approach the problem from a Bayesian inference perspective. We establish a probabilistic model for the problem and develop an improved turbo message passing (ITMP) algorithm based on the sum-product rule and the appropriate approximations. Additionally, we establish a state evolution framework to characterize the asymptotic behavior of the ITMP algorithm in the large-system limit. By analyzing the established state evolution, we further propose sufficient conditions for the global convergence of our algorithm. Our numerical results validate the theoretical results, demonstrating that the proposed asymptotic framework accurately characterize the dynamical behavior of the ITMP algorithm, and the phase transition curve specified by the sufficient condition agrees well with numerical simulations.