Approximate Message Passing for Multi-Layer Estimation in Rotationally Invariant Models

TL;DR

This paper introduces a new approximate message passing algorithm for multi-layer models with rotationally invariant weights, offering a computationally efficient alternative with performance comparable to existing methods.

Contribution

The paper proposes ML-RI-GAMP, a novel AMP algorithm for multi-layer rotationally invariant models that generalizes Gaussian AMP and reduces computational complexity.

Findings

ML-RI-GAMP accurately predicts performance via state evolution.

The algorithm achieves similar accuracy to ML-VAMP with lower complexity.

Numerical results confirm minimal performance loss despite reduced computational cost.

Abstract

We consider the problem of reconstructing the signal and the hidden variables from observations coming from a multi-layer network with rotationally invariant weight matrices. The multi-layer structure models inference from deep generative priors, and the rotational invariance imposed on the weights generalizes the i.i.d.\ Gaussian assumption by allowing for a complex correlation structure, which is typical in applications. In this work, we present a new class of approximate message passing (AMP) algorithms and give a state evolution recursion which precisely characterizes their performance in the large system limit. In contrast with the existing multi-layer VAMP (ML-VAMP) approach, our proposed AMP -- dubbed multi-layer rotationally invariant generalized AMP (ML-RI-GAMP) -- provides a natural generalization beyond Gaussian designs, in the sense that it recovers the existing Gaussian AMP as a special case. Furthermore, ML-RI-GAMP exhibits a significantly lower complexity than ML-VAMP, as the computationally intensive singular value decomposition is replaced by an estimation of the moments of the design matrices. Finally, our numerical results show that this complexity gain comes at little to no cost in the performance of the algorithm.