A Topological Approach to Spectral Clustering

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.