Phase Retrieval Under a Generative Prior

Abstract

The phase retrieval problem asks to recover a natural signal y0 ∈ R n from m quadratic observations, where m is to be minimized. As is common in many imaging problems, natural signals are considered sparse with respect to a known basis, and the generic sparsity prior is enforced via 1 regularization. While successful in the realm of linear inverse problems, such 1 methods have encountered possibly fundamental limitations, as no computationally efficient algorithm for phase retrieval of a k-sparse signal has been proven to succeed with fewer than O(k 2 log n) generic measurements, exceeding the theoretical optimum of O(k log n). In this paper, we propose a novel framework for phase retrieval by 1) modeling natural signals as being in the range of a deep generative neural network G : R k → R n and 2) enforcing this prior directly by optimizing an empirical risk objective over the domain of the generator. Our formulation has provably favorable global geometry for gradient methods, as soon as m = O(kd 2 log n), where d is the depth of the network. Specifically, when suitable deterministic conditions on the generator and measurement matrix are met, we construct a descent direction for any point outside of a small neighborhood around the unique global minimizer and its negative multiple, and show that such conditions hold with high probability under Gaussian ensembles of multilayer fully-connected generator networks and measurement matrices. This formulation for structured phase retrieval thus has two advantages over sparsity based methods: 1) deep generative priors can more tightly represent natural signals and 2) information theoretically optimal sample complexity. We corroborate these results with experiments showing that exploiting generative models in phase retrieval tasks outperforms sparse phase retrieval methods.

Robust Compressive Phase Retrieval via Deep Generative Priors

“…In final stages of this work we came to know about work of [48] that solves the phase retrieval problem using generative priors with rigorous theoretical guarantees for random Gaussian measurement matrix. We on the other hand provide rigorous experimental evaluation.…”

Just Least Squares: Binary Compressive Sampling with Low Generative Intrinsic Dimension

“…Proof. Similar as (19) in the proof to Theorem 2.1, by ( 26) in Lemma 2.5 and triangle inequality, we have with probability at least 0.99 that…”

“…While training, this mapping is encouraged to produce vectors that resemble the vectors in the training dataset. With this generative prior, several tasks have been studied such as image restoration [46], phase retrieval [19] and compressed sensing [53,4,23,35] and nonlinear single index models under certain measurement and noise models [52,34].…”

Deep Neural Networks are Surprisingly Reversible: A Baseline for Zero-Shot Inversion

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