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We consider the problem of compressed sensing and of (real-valued) phaseretrieval with random measurement matrix. We derive sharp asymptotics for theinformation-theoretically optimal performance and for the best known polynomialalgorithm for an ensemble of generative priors consisting of fully connecteddeep neural networks with random weight matrices and arbitrary activations. Wecompare the performance to sparse separable priors and conclude that generativepriors might be advantageous in terms of algorithmic performance. Inparticular, while sparsity does not allow to perform compressive phaseretrieval efficiently close to its information-theoretic limit, it is foundthat under the random generative prior compressed phase retrieval becomestractable.

Exact asymptotics for phase retrieval and compressed sensing with random generative priors