An Improved and Generalised Analysis for Spectral Clustering

TL;DR

This paper provides a more general theoretical analysis of Spectral Clustering, showing it works well with well-separated eigenvalues and extending its applicability to directed graphs and hierarchical clusters.

Contribution

It introduces a new, more general spectral analysis framework that applies to Hermitian representations of digraphs and hierarchical clustering regimes.

Findings

Spectral Clustering performs well with well-separated eigenvalues.

The analysis applies to directed graphs and hierarchical structures.

Results match empirical performance on synthetic and real data.

Abstract

We revisit the theoretical performances of Spectral Clustering, a classical algorithm for graph partitioning that relies on the eigenvectors of a matrix representation of the graph. Informally, we show that Spectral Clustering works well as long as the smallest eigenvalues appear in groups well separated from the rest of the matrix representation's spectrum. This arises, for example, whenever there exists a hierarchy of clusters at different scales, a regime not captured by previous analyses. Our results are very general and can be applied beyond the traditional graph Laplacian. In particular, we study Hermitian representations of digraphs and show Spectral Clustering can recover partitions where edges between clusters are oriented mostly in the same direction. This has applications in, for example, the analysis of trophic levels in ecological networks. We demonstrate that our results accurately predict the performances of Spectral Clustering on synthetic and real-world data sets.