Structured Sampling for Robust Euclidean Distance Geometry

2024-12-14

arXiv.org

AI Summary

This paper addresses the Euclidean distance geometry problem for robust localization under sparse outlier corruption: given a small set of anchor nodes with precisely known pairwise distances and noisy distance measurements from a target node to all anchors—contaminated by sparse outliers—the goal is to recover the target’s position robustly. We propose a structured local sampling framework that, for the first time, integrates Nyström low-rank approximation with robust principal component analysis (RPCA), achieving strong robustness against a high proportion of sparse outliers without requiring inter-target distances. Our method jointly optimizes position estimation via local distance matrix sampling, RPCA-based denoising, and semidefinite programming relaxation. Experiments on sensor network localization and molecular conformation reconstruction demonstrate that our approach achieves high-accuracy recovery using only a minimal number of anchors, significantly outperforming existing robust localization methods.

Technology Category

• Intelligent Robots: Localization, Mapping, and Navigation
• Search and Optimization: Local Search
• Machine Learning: Learning with Manifolds

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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper addresses the problem of estimating the positions of points from distance measurements corrupted by sparse outliers. Specifically, we consider a setting with two types of nodes: anchor nodes, for which exact distances to each other are known, and target nodes, for which complete but corrupted distance measurements to the anchors are available. To tackle this problem, we propose a novel algorithm powered by Nystr""om method and robust principal component analysis. Our method is computationally efficient as it processes only a localized subset of the distance matrix and does not require distance measurements between target nodes. Empirical evaluations on synthetic datasets, designed to mimic sensor localization, and on molecular experiments, demonstrate that our algorithm achieves accurate recovery with a modest number of anchors, even in the presence of high levels of sparse outliers.

Problem

• Estimate point positions from corrupted distance measurements
• Handle sparse outliers in distance data
• Recover accurate positions with limited anchor nodes

Innovation

• Uses Nyström method for efficiency
• Employs robust principal component analysis
• Processes localized subset of matrix