A Topological Approach to Spectral Clustering

TL;DR

This paper introduces two novel unsupervised clustering algorithms that leverage topological models and graph Laplacians to identify connected components in data sampled from metric spaces, without needing prior cluster information.

Contribution

The paper presents a topological approach to spectral clustering using graph Laplacians, with algorithms that do not require pre-specified cluster counts or auxiliary data.

Findings

Algorithms effectively identify connected components in data.

Approach works without prior knowledge of the number of clusters.

Utilizes topological models and graph Laplacians for clustering.

Abstract

We propose two related unsupervised clustering algorithms which, for input, take data assumed to be sampled from a uniform distribution supported on a metric space $X$, and output a clustering of the data based on the selection of a topological model for the connected components of $X$. Both algorithms work by selecting a graph on the samples from a natural one-parameter family of graphs, using a geometric criterion in the first case and an information theoretic criterion in the second. The estimated connected components of $X$ are identified with the kernel of the associated graph Laplacian, which allows the algorithm to work without requiring the number of expected clusters or other auxiliary data as input.