Data spectroscopy: Eigenspaces of convolution operators and clustering

TL;DR

This paper introduces a theoretical framework and a new clustering algorithm, DaSpec, based on eigenvectors of data adjacency matrices, improving cluster detection especially in unbalanced and complex-shaped data.

Contribution

The paper develops population analyses for eigenvector selection in spectral clustering and introduces DaSpec, a novel algorithm that automatically determines the number of clusters and improves clustering accuracy.

Findings

DaSpec effectively handles unbalanced groups.

It recovers clusters of various shapes better than existing methods.

Theoretical insights explain when and why spectral methods succeed or fail.

Abstract

This paper focuses on obtaining clustering information about a distribution from its i.i.d. samples. We develop theoretical results to understand and use clustering information contained in the eigenvectors of data adjacency matrices based on a radial kernel function with a sufficiently fast tail decay. In particular, we provide population analyses to gain insights into which eigenvectors should be used and when the clustering information for the distribution can be recovered from the sample. We learn that a fixed number of top eigenvectors might at the same time contain redundant clustering information and miss relevant clustering information. We use this insight to design the data spectroscopic clustering (DaSpec) algorithm that utilizes properly selected eigenvectors to determine the number of clusters automatically and to group the data accordingly. Our findings extend the intuitions underlying existing spectral techniques such as spectral clustering and Kernel Principal Components Analysis, and provide new understanding into their usability and modes of failure. Simulation studies and experiments on real-world data are conducted to show the potential of our algorithm. In particular, DaSpec is found to handle unbalanced groups and recover clusters of different shapes better than the competing methods.