  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.
