A coherence parameter characterizing generative compressed sensing with Fourier measurements

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 with nearly order-optimal sample complexity, 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.