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.

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Employing deep neural networks as natural image priors to solve inverse problems either requires large amounts of data to sufficiently train expressive generative models or can succeed with no data via untrained neural networks. However, very few works have considered how to interpolate between these no-to high-data regimes. In particular, how can one use the availability of a small amount of data (even 5 − 25 examples) to one's advantage in solving these inverse problems and can a system's performance increase as the amount of data increases as well? In this work, we consider solving linear inverse problems when given a small number of examples of images that are drawn from the same distribution as the image of interest. Comparing to untrained neural networks that use no data, we show how one can pre-train a neural network with a few given examples to improve reconstruction results in compressed sensing and semantic image recovery problems such as colorization. Our approach leads to improved reconstruction as the amount of available data increases and is on par with fully trained generative models, while requiring less than 1% of the data needed to train a generative model.

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