Macroscopic Analysis of Vector Approximate Message Passing in a Model Mismatch Setting

TL;DR

This paper extends the analysis of VAMP to model mismatch scenarios, deriving state evolution equations and revealing conditions for stability and replica symmetry breaking, supported by numerical experiments.

Contribution

It provides the first detailed analysis of VAMP performance under model mismatch, including stability conditions and connections to replica symmetry breaking.

Findings

State evolution equations describe VAMP dynamics under mismatch.

Fixed point analysis reveals conditions for replica symmetry breaking.

Numerical results support theoretical predictions.

Abstract

Vector approximate message passing (VAMP) is an efficient approximate inference algorithm used for generalized linear models. Although VAMP exhibits excellent performance, particularly when measurement matrices are sampled from rotationally invariant ensembles, existing convergence and performance analyses have been limited mostly to cases in which the correct posterior distribution is available. Here, we extend the analyses for cases in which the correct posterior distribution is not used in the inference stage. We derive state evolution equations, which macroscopically describe the dynamics of VAMP, and show that their fixed point is consistent with the replica symmetric solution obtained by the replica method of statistical mechanics. We also show that the fixed point of VAMP can exhibit a microscopic instability, the critical condition of which agrees with that for breaking the replica symmetry. The results of numerical experiments support our findings.