Abstract:We study nonconvex optimization for phase retrieval and the more general problem of semidefinite low-rank matrix sensing; in particular, we focus on the global nonconvex landscape of overparametrized versions of the nonsmooth amplitude least-squares loss as well as a smooth reformulation of this loss based on the PhaseCut approach. We first give a general, deterministic result on properties of second-order critical points for a general class of loss functions; we then specialize this result to the nonsmooth amplitude loss and, additionally, prove nearly identical results for a smooth reformulation (similar to PhaseCut) as a synchronization problem over spheres. Finally, we show the usefulness of these tools by proving high-probability landscape guarantees in two settings: (1) phase retrieval with isotropic sub-Gaussian measurements, and (2) phase retrieval in a general (possibly infinite-dimensional) Hilbert space with Gaussian measurements. In both cases, our results give state-of-the-art and statistically optimal guarantees with only a constant amount of overparametrization (in the well-studied case of isotropic sub-Gaussian measurements, such statistical guarantees had previously required greater degrees of overparametrization/relaxation); this demonstrates the potential of overparametrized nonconvex optimization as a principled and scalable algorithmic approach to phase retrieval.
From: Andrew McRae [view email]
[v1]
Mon, 24 Nov 2025 12:33:01 UTC (35 KB)
[v2]
Mon, 29 Jun 2026 12:43:35 UTC (38 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。