Misspecified Phase Retrieval with Generative Priors
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In this paper, we study phase retrieval under model misspecification and generative priors. In particular, we aim to estimate an n-dimensional signal ๐ฑ from m i.i.d.ย realizations of the single index model y = f(๐^T๐ฑ), where f is an unknown and possibly random nonlinear link function and ๐โโ^n is a standard Gaussian vector. We make the assumption Cov[y,(๐^T๐ฑ)^2] 0, which corresponds to the misspecified phase retrieval problem. In addition, the underlying signal ๐ฑ is assumed to lie in the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. We propose a two-step approach, for which the first step plays the role of spectral initialization and the second step refines the estimated vector produced by the first step iteratively. We show that both steps enjoy a statistical rate of order โ((klog L)ยท (log m)/m) under suitable conditions. Experiments on image datasets are performed to demonstrate that our approach performs on par with or even significantly outperforms several competing methods.
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