Weighted Riemannian Optimization for Solving Quadratic Equations from Gaussian Magnitude Measurements

2026-04-15

AI Summary

This work addresses the generalized phase retrieval problem under Gaussian amplitude measurements, aiming to reconstruct an $ n $-dimensional signal from phaseless quadratic observations. The problem is reformulated as a least-squares fit of linear equations over the manifold of rank-1 positive semidefinite matrices. To enhance the restricted isometry property of the measurement operator on this manifold, the authors introduce a novel weighted Riemannian metric. Building upon this geometric insight, they propose a Weighted Riemannian Gradient Descent (WRGD) algorithm coupled with spectral initialization. Theoretical analysis establishes that WRGD enjoys linear convergence with an improved contraction factor compared to existing methods. Numerical experiments demonstrate that WRGD outperforms both Truncated Wirtinger Flow (TWF) and standard Riemannian Gradient Descent in terms of computational efficiency and robustness.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

This paper explores the problem of generalized phase retrieval, which involves reconstructing a length-$n$ signal $\bm{x}$ from its $m$ phaseless samples $y_k = \left|\langle \bm{a}_k,\bm{x}\rangle\right|^2$, where $k = 1,2,...,m$, and $\bm{a}_k$ are the measurement vectors. This problem can be reformulated into recovering a positive semidefinite rank-$1$ matrix $\bm{X}=\bm{x}\bm{x}^*$ from linear samples $\bm{y}=\mathcal{A}(\bm{X})\in\mathbb{R}^m$, thereby requiring us to find a rank-$1$ solution of the linear equations. We demonstrate that several existing phase retrieval algorithms, including Wirtinger Flow (WF) and the canonical Riemannian gradient descent (RGD), actually solve the least-squares fitting of this linear equation on the Riemannian manifold of rank-$1$ matrices, but utilize different metrics on this manifold. Nevertheless, these metrics only allow for a stable and far-apart-from-isometric embedding of rank-$1$ matrices to $\mathbb{R}^m$ by $\mathcal{A}$, resulting in a linear convergence with a considerably large convergence factor. To expedite the convergence, we establish a new metric on the rank-$1$ matrix manifold that facilitates the nearly isometric embedding of rank-$1$ matrices into $\mathbb{R}^m$ through $\mathcal{A}$. A RGD algorithm under this new metric, termed Weighted RGD (WRGD), is proposed to tackle the phase retrieval problem. Owing to the near isometry, we prove that our WRGD algorithm, initialized by spectral methods, can linearly converge to the underlying signal $\bm{x}$ with a small convergence factor. Empirical experiments strongly validate the efficiency and resilience of our algorithms compared to the truncated Wirtinger Flow (TWF) algorithm and the canonical RGD algorithm.

Problem

• phase retrieval
• rank-1 matrix recovery
• Gaussian magnitude measurements
• quadratic equations
• positive semidefinite matrix

Innovation

• Weighted Riemannian Optimization
• Phase Retrieval
• Nearly Isometric Embedding
• Rank-1 Matrix Manifold
• Linear Convergence