Multi-layer state evolution under random convolutional design
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.
WOS:001105355700001
2023-11-01
2023
11
114002
REVIEWED
Funder | Grant Number |
M D acknowledges funding from Northeastern University's Undergraduate Research amp; Fellowships office and the Goldwater Award. We acknowledge funding from the ERC under the European Union's Horizon 2020 Research and Innovation Program Grant Agreement 714 | |
Northeastern University's Undergraduate Research amp; Fellowships office and the Goldwater Award | 714608-SMiLe |
ERC under the European Union | |