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.

翻译：紧致群上的广义相位恢复问题旨在从关联的格拉姆矩阵中恢复一组表示未知信号的矩阵。该框架将经典的相位恢复问题——即从傅里叶变换的幅度中重构信号——推广至涉及非阿贝尔紧致群的更丰富场景。在这一更广泛的背景下，傅里叶空间中的未知相位被替换为由紧致群在有限维向量空间上的作用所产生的未知正交矩阵。该问题主要受到电子显微镜领域进展的推动，旨在从高度噪声的观测中确定生物大分子的三维结构。为捕捉机器学习和信号处理中的实际假设，我们将信号建模为属于若干广泛结构族之一：一个通用线性子空间、一个通用基下的稀疏表示、一个通用ReLU神经网络的输出，或一个通用低维流形。我们的主要结果表明，对于足够低维的先验，广义相位恢复问题不仅允许唯一解（直至固有的群对称性），而且满足双利普希茨性质。这意味着对噪声和模型失配均具有鲁棒性——这是实际应用中的基本要求，尤其是在测量被噪声严重破坏的情况下。这些发现为现代结构假设下的一大类科学问题提供了理论支持，并为在高噪声环境下开发鲁棒算法奠定了坚实基础。