Provable Non-Convex Euclidean Distance Matrix Completion: Geometry, Reconstruction, and Robustness

TL;DR

This paper introduces a Riemannian optimization approach for Euclidean Distance Matrix Completion, providing theoretical guarantees, a novel initialization method, and empirical validation for recovering point configurations from partial distances.

Contribution

It formulates EDMC as a low-rank matrix completion problem on a Riemannian manifold, with new convergence analysis, initialization strategy, and robustness guarantees.

Findings

Linear convergence of Riemannian gradient descent under certain sampling conditions

A one-step thresholding initialization ensures convergence with high probability

Empirical results show competitive performance with existing methods

Abstract

The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) 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 EDMC 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 nonnegativity and 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 $p\geq O(\nu^2 r^2\log(n)/n)$, where $\nu$ is an EDMC-specific incoherence parameter. Furthermore, we provide an initialization candidate using a one-step hard thresholding procedure that yields convergence, provided the sampling probability satisfies $p \geq O(\nu r^{3/2}\log^{3/4}(n)/n^{1/4})$. 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 analysis of a second order degenerate U-statistic 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 provide a geometric interpretation of matrix incoherence tailored to the EDMC setting and provide robustness guarantees for our method.