A coherence parameter characterizing generative compressed sensing with Fourier measurements

TL;DR

This paper introduces a new coherence parameter to characterize the effectiveness of generative compressed sensing with Fourier measurements, providing theoretical guarantees and a training strategy for improved recovery.

Contribution

It extends compressed sensing theory to subsampled isometries, introduces the coherence parameter, and proposes a regularization method for training neural networks with favorable coherence.

Findings

Proves the first restricted isometry guarantee for subsampled isometries in generative compressed sensing.

Shows that low coherence networks require fewer measurements for accurate recovery.

Provides numerical evidence supporting the regularization strategy for training neural networks with low coherence.

Abstract

In Bora et al. (2017), a mathematical framework was developed for compressed sensing guarantees in the setting where the measurement matrix is Gaussian and the signal structure is the range of a generative neural network (GNN). The problem of compressed sensing with GNNs has since been extensively analyzed when the measurement matrix and/or network weights follow a subgaussian distribution. We move beyond the subgaussian assumption, to measurement matrices that are derived by sampling uniformly at random rows of a unitary matrix (including subsampled Fourier measurements as a special case). Specifically, we prove the first known restricted isometry guarantee for generative compressed sensing with subsampled isometries and provide recovery bounds, addressing an open problem of Scarlett et al. (2022, p. 10). Recovery efficacy is characterized by the coherence, a new parameter, which measures the interplay between the range of the network and the measurement matrix. Our approach relies on subspace counting arguments and ideas central to high-dimensional probability. Furthermore, we propose a regularization strategy for training GNNs to have favourable coherence with the measurement operator. We provide compelling numerical simulations that support this regularized training strategy: our strategy yields low coherence networks that require fewer measurements for signal recovery. This, together with our theoretical results, supports coherence as a natural quantity for characterizing generative compressed sensing with subsampled isometries.