Multi-layer State Evolution Under Random Convolutional Design

TL;DR

This paper extends the theoretical analysis of signal recovery algorithms to convolutional neural network priors, demonstrating that random convolutional layers behave similarly to Gaussian matrices, which broadens the applicability of state evolution methods.

Contribution

It establishes the state evolution of ML-AMP for random convolutional layers, showing their universality class equivalence to Gaussian matrices, and introduces a novel proof technique linking convolutional matrices to coding theory.

Findings

Random convolutional layers belong to the same universality class as Gaussian matrices.

The proof technique maps convolutional matrices to spatially coupled sensing matrices.

The results enable rigorous analysis of convolutional neural network priors in signal recovery.

Abstract

Signal recovery under generative neural network priors has emerged as a promising direction in statistical inference and computational imaging. Theoretical analysis of reconstruction algorithms under generative priors is, however, challenging. For generative priors with fully connected layers and Gaussian i.i.d. weights, this was achieved by the multi-layer approximate message (ML-AMP) algorithm via a rigorous state evolution. However, practical generative priors are typically convolutional, allowing for computational benefits and inductive biases, and so the Gaussian i.i.d. weight assumption is very limiting. In this paper, we overcome this limitation and establish the state evolution of ML-AMP for random convolutional layers. We prove in particular that random convolutional layers belong to the same universality class as Gaussian matrices. Our proof technique is of an independent interest as it establishes a mapping between convolutional matrices and spatially coupled sensing matrices used in coding theory.