Vector Approximate Message Passing

TL;DR

This paper introduces the Vector AMP (VAMP) algorithm, which extends the applicability of approximate message passing to a broader class of matrices, providing rigorous state-evolution analysis and matching theoretical MMSE predictions.

Contribution

The paper develops VAMP with a rigorous state-evolution for right-orthogonally invariant matrices, broadening AMP's scope and confirming its fixed points align with replica predictions.

Findings

VAMP's state evolution is valid for right-orthogonally invariant matrices.

VAMP's fixed points match the replica minimum mean-squared error predictions.

Numerical results confirm VAMP's effectiveness and theoretical predictions.

Abstract

The standard linear regression (SLR) problem is to recover a vector $\mathbf{x}^0$ from noisy linear observations $\mathbf{y}=\mathbf{Ax}^0+\mathbf{w}$. The approximate message passing (AMP) algorithm recently proposed by Donoho, Maleki, and Montanari is a computationally efficient iterative approach to SLR that has a remarkable property: for large i.i.d.\ sub-Gaussian matrices $\mathbf{A}$, its per-iteration behavior is rigorously characterized by a scalar state-evolution whose fixed points, when unique, are Bayes optimal. The AMP algorithm, however, is fragile in that even small deviations from the i.i.d.\ sub-Gaussian model can cause the algorithm to diverge. This paper considers a "vector AMP" (VAMP) algorithm and shows that VAMP has a rigorous scalar state-evolution that holds under a much broader class of large random matrices $\mathbf{A}$: those that are right-orthogonally invariant. After performing an initial singular value decomposition (SVD) of $\mathbf{A}$, the per-iteration complexity of VAMP can be made similar to that of AMP. In addition, the fixed points of VAMP's state evolution are consistent with the replica prediction of the minimum mean-squared error recently derived by Tulino, Caire, Verd\'u, and Shamai. Numerical experiments are used to confirm the effectiveness of VAMP and its consistency with state-evolution predictions.