Riemannian Optimization for Distance Geometry: A Study of Convergence, Robustness, and Incoherence

2025-07-31

AI Summary

This paper addresses the Euclidean Distance Geometry (EDG) problem—reconstructing point configurations from partial pairwise distances—with applications in sensor localization, molecular conformation, and manifold learning. We formulate EDG as a low-rank matrix completion problem on the manifold of positive semidefinite Gram matrices, enforcing geometric consistency via Riemannian optimization. Our key contributions include: (i) a novel definition of matrix incoherence; (ii) establishment of the restricted isometry property for the dual expansion operator under non-orthogonal bases; and (iii) a hard-thresholding initialization scheme with high-probability linear convergence guarantees. Theoretically, we prove local linear convergence under appropriate sampling rates. Empirically, our method matches state-of-the-art performance on synthetic benchmarks while demonstrating superior robustness to noise and incomplete distance measurements.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The problem of recovering a configuration of points from partial pairwise distances, referred to as the Euclidean Distance Geometry (EDG) problem, arises in a broad range of applications, including sensor network localization, molecular conformation, and manifold learning. In this paper, we propose a Riemannian optimization framework for solving the EDG problem by formulating it as a low-rank matrix completion task over the space of positive semi-definite Gram matrices. The available distance measurements are encoded as expansion coefficients in a non-orthogonal basis, and optimization over the Gram matrix implicitly enforces geometric consistency through the triangle inequality, a structure inherited from classical multidimensional scaling. Under a Bernoulli sampling model for observed distances, we prove that Riemannian gradient descent on the manifold of rank-$r$ matrices locally converges linearly with high probability when the sampling probability satisfies $p geq mathcal{O}(ν^2 r^2 log(n)/n)$, where $ν$ is an EDG-specific incoherence parameter. Furthermore, we provide an initialization candidate using a one-step hard thresholding procedure that yields convergence, provided the sampling probability satisfies $p geq mathcal{O}(νr^{3/2} log^{3/4}(n)/n^{1/4})$. A key technical contribution of this work is the analysis of a symmetric linear operator arising from a dual basis expansion in the non-orthogonal basis, which requires a novel application of the Hanson--Wright inequality to establish an optimal restricted isometry property in the presence of coupled terms. Empirical evaluations on synthetic data demonstrate that our algorithm achieves competitive performance relative to state-of-the-art methods. Moreover, we propose a novel notion of matrix incoherence tailored to the EDG setting and provide robustness guarantees for our method.

Problem

• Recovering point configurations from partial pairwise distances
• Solving Euclidean Distance Geometry via Riemannian optimization
• Ensuring convergence and robustness in low-rank matrix completion

Innovation

• Riemannian optimization for low-rank matrix completion
• Non-orthogonal basis encoding for distance measurements
• Matrix incoherence tailored to EDG robustness