The stability of generalized phase retrieval problem over compact groups

2025-05-07

AI Summary

This work addresses the generalized phase retrieval problem on compact groups, aiming to robustly reconstruct an unknown matrix signal—endowed with structural priors such as subspace membership, sparsity, ReLU-network outputs, or low-dimensional manifold structure—from noisy Gram matrices. Extending classical phase retrieval, it replaces Fourier magnitudes with unknown orthogonal matrices arising from non-Abelian group actions, directly motivated by high-noise 3D structural reconstruction in single-particle cryo-electron microscopy. Methodologically, the approach integrates harmonic analysis, representation theory of matrix groups, nonlinear compressed sensing, and structured signal modeling. The key contribution is the first double-Lipschitz stability theory for generalized phase retrieval on compact groups, rigorously establishing uniqueness of solutions modulo group symmetries and strong robustness to both measurement noise and model mismatch. This provides the first theoretically guaranteed, robust reconstruction framework for biomolecular structure determination.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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, leveraging prior structural knowledge about the signal. 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, under mild conditions, 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.

Problem

• Recovering signal matrices from Gram matrices using compact groups
• Generalizing phase retrieval to non-abelian groups and noisy data
• Ensuring solution uniqueness and robustness under structural constraints

Innovation

• Recover matrices from Gram matrices using compact groups
• Model signals with structural families like neural networks
• Ensure robustness to noise with bi-Lipschitz property