A Dual Basis Approach for Structured Robust Euclidean Distance Geometry

2025-05-23

AI Summary

This paper addresses the problem of robust point configuration recovery from structured, partially observed Euclidean distance matrices (EDMs) under anchor-node scarcity and outlier corruption. The proposed method introduces, for the first time, non-orthogonal dual bases—thereby relaxing the conventional assumptions of orthogonality and full observability—and establishes theoretical guarantees for the joint exact recovery of both the Gram matrix and the underlying point configuration. Methodologically, it integrates robust low-rank matrix recovery, structured sparsity modeling, and semidefinite programming relaxation into a unified optimization framework. Evaluated on sensor localization and molecular conformation datasets, the approach significantly outperforms state-of-the-art methods. Under theoretically grounded conditions, it achieves exact reconstruction of both the Gram matrix and the point configuration, effectively resolving the robust localization challenge posed by sparse and corrupted EDMs.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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.

Problem

• Recovering Euclidean distance geometry from partial corrupted observations
• Handling outliers in structured partial Euclidean Distance Matrix data
• Developing a dual basis approach for robust distance geometry recovery

Innovation

• Uses dual basis for robust EDM recovery
• Handles partial corruptions in distance data
• Ensures exact recovery of Gram matrix