A topological approach to spectral clustering

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula>, and output a clustering of the data based on the selection of a topological model for the connected components of <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline-formula>. Both algorithms work by selecting a graph on the samples from a natural one-parameter family of graphs, using a geometric criterion in the first case and an information theoretic criterion in the second. The estimated connected components of <inline-formula><tex-math id="M3">\begin{document}$ X $\end{document}</tex-math></inline-formula> 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.