Consistent Spectral Clustering in Hyperbolic Spaces

TL;DR

This paper introduces a spectral clustering algorithm designed for hyperbolic spaces, which better captures complex hierarchical data structures and demonstrates improved efficiency and consistency over traditional Euclidean spectral clustering.

Contribution

The paper develops a novel spectral clustering algorithm for hyperbolic spaces and proves its weak consistency, advancing clustering methods for non-Euclidean data representations.

Findings

Hyperbolic spectral clustering outperforms Euclidean methods on complex data.

The algorithm converges at least as fast as Euclidean spectral clustering.

Experimental results on breast cancer data validate the approach.

Abstract

Clustering, as an unsupervised technique, plays a pivotal role in various data analysis applications. Among clustering algorithms, Spectral Clustering on Euclidean Spaces has been extensively studied. However, with the rapid evolution of data complexity, Euclidean Space is proving to be inefficient for representing and learning algorithms. Although Deep Neural Networks on hyperbolic spaces have gained recent traction, clustering algorithms or non-deep machine learning models on non-Euclidean Spaces remain underexplored. In this paper, we propose a spectral clustering algorithm on Hyperbolic Spaces to address this gap. Hyperbolic Spaces offer advantages in representing complex data structures like hierarchical and tree-like structures, which cannot be embedded efficiently in Euclidean Spaces. Our proposed algorithm replaces the Euclidean Similarity Matrix with an appropriate Hyperbolic Similarity Matrix, demonstrating improved efficiency compared to clustering in Euclidean Spaces. Our contributions include the development of the spectral clustering algorithm on Hyperbolic Spaces and the proof of its weak consistency. We show that our algorithm converges at least as fast as Spectral Clustering on Euclidean Spaces. To illustrate the efficacy of our approach, we present experimental results on the Wisconsin Breast Cancer Dataset, highlighting the superior performance of Hyperbolic Spectral Clustering over its Euclidean counterpart. This work opens up avenues for utilizing non-Euclidean Spaces in clustering algorithms, offering new perspectives for handling complex data structures and improving clustering efficiency.