A Dual Basis Approach for Structured Robust Euclidean Distance Geometry

TL;DR

This paper introduces RoDEoDB, a novel algorithm leveraging dual basis techniques to robustly recover Euclidean distance geometry from partial, potentially corrupted distance data, with proven guarantees and superior empirical performance.

Contribution

It proposes a new dual basis-based framework for robust Euclidean distance geometry recovery, addressing partial observations and outliers with theoretical guarantees.

Findings

Exact recovery guarantees under mild conditions

Superior performance on sensor localization datasets

Effective handling of outliers in distance data

Abstract

Euclidean Distance Matrix (EDM), which consists of pairwise squared Euclidean distances of a given point configuration, finds many applications in modern machine learning. This paper considers the setting where only a set of anchor nodes is used to collect the distances between themselves and the rest. In the presence of potential outliers, it results in a structured partial observation on EDM with partial corruptions. Note that an EDM can be connected to a positive semi-definite Gram matrix via a non-orthogonal dual basis. Inspired by recent development of non-orthogonal dual basis in optimization, we propose a novel algorithmic framework, dubbed Robust Euclidean Distance Geometry via Dual Basis (RoDEoDB), for recovering the Euclidean distance geometry, i.e., the underlying point configuration. The exact recovery guarantees have been established in terms of both the Gram matrix and point configuration, under some mild conditions. Empirical experiments show superior performance of RoDEoDB on sensor localization and molecular conformation datasets.