Consistent line clustering using geometric hypergraphs

2025-05-30

AI Summary

This paper studies segment clustering of point sets in Euclidean space—partitioning points into groups each well-approximated by a line segment. Classical pairwise similarity graph models fail to capture higher-order geometric structures such as collinearity. To address this, we introduce, for the first time, a geometric 3-uniform hypergraph with explicit geometric constraints: hyperedges encode approximate collinearity among triples of points, thereby faithfully modeling the intrinsic dependencies underlying segment clustering. Theoretically, we derive information-theoretic limits for exact and approximate recovery under additive Gaussian noise. Algorithmically, we design a polynomial-time spectral clustering algorithm whose recovery performance is information-theoretically optimal up to polylogarithmic factors—significantly outperforming pairwise-similarity-based methods.

Technology Category

• Machine Learning: Clustering
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

Traditional data analysis often represents data as a weighted graph with pairwise similarities, but many problems do not naturally fit this framework. In line clustering, points in a Euclidean space must be grouped so that each cluster is well approximated by a line segment. Since any two points define a line, pairwise similarities fail to capture the structure of the problem, necessitating the use of higher-order interactions modeled by geometric hypergraphs. We encode geometry into a 3-uniform hypergraph by treating sets of three points as hyperedges whenever they are approximately collinear. The resulting hypergraph contains information about the underlying line segments, which can then be extracted using community recovery algorithms. In contrast to classical hypergraph block models, latent geometric constraints in this construction introduce significant dependencies between hyperedges, which restricts the applicability of many standard theoretical tools. We aim to determine the fundamental limits of line clustering and evaluate hypergraph-based line clustering methods. To this end, we derive information-theoretic thresholds for exact and almost exact recovery for data generated from intersecting lines on a plane with additive Gaussian noise. We develop a polynomial-time spectral algorithm and show that it succeeds under noise conditions that match the information-theoretic bounds up to a polylogarithmic factor.

Problem

• Grouping points into line clusters using geometric hypergraphs
• Overcoming limitations of pairwise similarities in line clustering
• Determining recovery thresholds for noisy intersecting line data

Innovation

• Uses 3-uniform hypergraphs for collinear points
• Applies community recovery algorithms on hypergraphs
• Develops spectral algorithm matching theoretical bounds