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

Abstract

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.

13+3 pages, 7 figures, v2 revised and accepted at MSML

References in corpus (4)

• Message Passing Algorithms for Compressed Sensing
• Entropy and mutual information in models of deep neural networks
• Phase Retrieval Under a Generative Prior
• SUNLayer: Stable denoising with generative networks

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• Deep Learning Techniques for Inverse Problems in Imaging
• Theoretical characterization of uncertainty in high-dimensional linear classification
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• Construction of optimal spectral methods in phase retrieval