A coherence parameter characterizing generative compressed sensing with Fourier measurements


We study compressed sensing when the signal structure is the range of a (pre-trained) generative neural network (GNN) and give the first signal recovery guarantees when the measurements are subsampled from an orthonormal basis. This includes the subsampled Fourier transform as a special case, thus allowing a measurement model in line with applications such as MRI. We define a coherence parameter which depends upon the alignment of the orthonormal basis and the range of the GNN and show that small coherence implies robust signal recovery from relatively few measurements. Numerical experiments verify that regularizing to keep the coherence parameter small while training the GNN can improve performance in the signal recovery stage.