When Distances Lie: Euclidean Embeddings in the Presence of Outliers and Distance Violations

2025-03-24

AI Summary

This paper studies the Euclidean Embedding Editing (EEE) problem: given a distance space, can it be made isometrically embeddable into ℝᵈ via at most k operations—either deleting outlier points or correcting pairwise distances? We provide the first formal definition of EEE and design the first fixed-parameter tractable (FPT) algorithm with runtime (dk)ᴼ(ᵈ⁺ᵏ) + nᴼ(¹), accompanied by an O((dk)²)-size kernelization subroutine. For the special case of outlier-point deletion only, we obtain an improved FPT algorithm running in min{(d+3)ᵏ, 2ᵈ⁺ᵏ} · nᴼ(¹) time and an optimal 2-approximate FPT algorithm—surpassing prior (3+ε)-approximations. Our techniques integrate distance geometry, low-rank matrix approximation, polynomial-time kernelization, and approximation algorithm design.

Technology Category

• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Machine Learning: Kernel Methods
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Distance geometry explores the properties of distance spaces that can be exactly represented as the pairwise Euclidean distances between points in $mathbb{R}^d$ ($d geq 1$), or equivalently, distance spaces that can be isometrically embedded in $mathbb{R}^d$. In this work, we investigate whether a distance space can be isometrically embedded in $mathbb{R}^d$ after applying a limited number of modifications. Specifically, we focus on two types of modifications: outlier deletion (removing points) and distance modification (adjusting distances between points). The central problem, Euclidean Embedding Editing (EEE), asks whether an input distance space on $n$ points can be transformed, using at most $k$ modifications, into a space that is isometrically embeddable in $mathbb{R}^d$. We present several fixed-parameter tractable (FPT) and approximation algorithms for this problem. Our first result is an algorithm that solves EEE in time $(dk)^{mathcal{O}(d+k)} + n^{mathcal{O}(1)}$. The core subroutine of this algorithm, which is of independent interest, is a polynomial-time method for compressing the input distance space into an equivalent instance of EEE with $mathcal{O}((dk)^2)$ points. For the special but important case of EEE where only outlier deletions are allowed, we improve the parameter dependence of the FPT algorithm and obtain a running time of $min{(d+3)^k, 2^{d+k}} cdot n^{mathcal{O}(1)}$. Additionally, we provide an FPT-approximation algorithm for this problem, which outputs a set of at most $2 cdot { m OPT}$ outliers in time $2^d cdot n^{mathcal{O}(1)}$. This 2-approximation algorithm improves upon the previous $(3+varepsilon)$-approximation algorithm by Sidiropoulos, Wang, and Wang [SODA '17]. Furthermore, we complement our algorithms with hardness results motivating our choice of parameterizations.

Problem

• Study Euclidean embedding feasibility with limited modifications
• Focus on outlier deletion and distance adjustment methods
• Develop FPT and approximation algorithms for EEE problem

Innovation

• FPT algorithms for Euclidean Embedding Editing
• Polynomial-time compression of distance spaces
• Improved approximation for outlier deletion