Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which $Y = f(\boldsymbol{X}^{\top}\boldsymbol{\beta}^*, \varepsilon)$ with unknown $f$ and $\operatorname{Cov}(Y, (\boldsymbol{X}^{\top}\boldsymbol{\beta}^*)^2) > 0$. In this paper, we propose an estimation procedure, which consists of solving a cascade of two convex programs and provably recovers the direction of $\boldsymbol{\beta}^*$.

We introduce a novel deep-learning inspired formulation of the \textit{phase retrieval problem}, which asks to recover a signal $y_0 \in \R^n$ from $m$ quadratic observations, under structural assumptions on the underlying signal. As is common in many imaging problems, previous methodologies have considered natural signals as being sparse with respect to a known basis, resulting in the decision to enforce a generic sparsity prior. However, these methods for phase retrieval have encountered possibly fundamental limitations, as no computationally efficient algorithm for sparse phase retrieval has been proven to succeed with fewer than $O(k^2\log n)$ generic measurements, which is larger than the theoretical optimum of $O(k \log n)$. In this paper, we sidestep this issue by considering a prior that a natural signal is in the range of a generative neural network $G: \R^k \rightarrow \R^n$. We introduce an empirical risk formulation that has favorable global geometry for gradient methods, as soon as $m = O(k)$, under the model of a multilayer fully-connected neural network with random weights. Specifically, we show that there exists a descent direction outside of a small neighborhood around the true $k$-dimensional latent code and a negative multiple thereof. This formulation for structured phase retrieval thus benefits from two effects: generative priors can more tightly represent natural signals than sparsity priors, and this empirical risk formulation can exploit those generative priors at an information theoretically optimal sample complexity, unlike for a sparsity prior. We corroborate these results with experiments showing that exploiting generative models in phase retrieval tasks outperforms both sparse and general phase retrieval methods.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Mirror descent [37] is becoming increasingly popular in a variety of settings in optimization and machine learning. One reason for its success is the fact that mirror descent can be adapted to fit the geometry ofthe optimization problem athand bychoosing asuitable strictly convexpotential function,theso-calledmirrormap.