The stability of generalized phase retrieval problem over compact groups
Tal Amir, Tamir Bendory, Nadav Dym, Dan Edidin
TL;DR
We study generalized phase retrieval over compact groups with the goal of recovering a signal $X \in V$ from its second-moment Gram matrices $X^{\top}X$. The main contribution shows that, under low-dimensional semi-algebraic priors such as linear subspaces, sparsity, generative ReLU networks, or low-dimensional manifolds, the map $X \mapsto \sqrt{X^{\top}X}$ is bi-Lipschitz up to sign when restricted to the prior, provided the prior dimension $M$ satisfies $2M < K$, where $K$ is the effective dimension under the group action. This yields uniqueness (up to sign) and robust stability to noise and model mismatch, with concrete corollaries for the priors and a manifold extension. The results have direct implications for cryo-EM and related MRA-like setups, offering stability guarantees and informing sample complexity in high-noise regimes for second-moment-based reconstruction methods.
Abstract
The generalized phase retrieval problem over compact groups aims to recover a set of matrices -- representing an unknown signal -- from their associated Gram matrices. This framework generalizes the classical phase retrieval problem, which reconstructs a signal from the magnitudes of its Fourier transform, to a richer setting involving non-abelian compact groups. In this broader context, the unknown phases in Fourier space are replaced by unknown orthogonal matrices that arise from the action of a compact group on a finite-dimensional vector space. This problem is primarily motivated by advances in electron microscopy to determining the 3D structure of biological macromolecules from highly noisy observations. To capture realistic assumptions from machine learning and signal processing, we model the signal as belonging to one of several broad structural families: a generic linear subspace, a sparse representation in a generic basis, the output of a generic ReLU neural network, or a generic low-dimensional manifold. Our main result shows that, for a prior of sufficiently low dimension, the generalized phase retrieval problem not only admits a unique solution (up to inherent group symmetries), but also satisfies a bi-Lipschitz property. This implies robustness to both noise and model mismatch -- an essential requirement for practical use, especially when measurements are severely corrupted by noise. These findings provide theoretical support for a wide class of scientific problems under modern structural assumptions, and they offer strong foundations for developing robust algorithms in high-noise regimes.