Exact asymptotics for phase retrieval and compressed sensing with random generative priors
This work addresses the challenge of efficient signal recovery in compressed sensing and phase retrieval for applications like imaging and signal processing, showing a potential advantage of generative models over sparsity, though it is incremental in extending prior asymptotic analyses.
The paper tackles the problem of compressed sensing and phase retrieval with random measurement matrices, deriving exact asymptotics for the information-theoretically optimal performance and best polynomial algorithm under random generative priors based on deep neural networks. It finds that generative priors make compressed phase retrieval tractable near its information-theoretic limit, unlike sparse priors.
We consider the problem of compressed sensing and of (real-valued) phase retrieval with random measurement matrix. We derive sharp asymptotics for the information-theoretically optimal performance and for the best known polynomial algorithm for an ensemble of generative priors consisting of fully connected deep neural networks with random weight matrices and arbitrary activations. We compare the performance to sparse separable priors and conclude that generative priors might be advantageous in terms of algorithmic performance. In particular, while sparsity does not allow to perform compressive phase retrieval efficiently close to its information-theoretic limit, it is found that under the random generative prior compressed phase retrieval becomes tractable.