Data Spectroscopy: Eigenspace Of Convolution Operators And Clustering
This paper focuses on obtaining clustering information in a distribution when iid data are given. First, we develop theoretical results for understanding and using clustering information contained in the eigenvectors of data adjacency matrices based on a radial kernel function (with a sufficiently fast tail decay). We provide population analyses to give insights into which eigenvectors should be used and when the clustering information for the distribution can be recovered from the data. In particular, we learned that top eigenvectors do not contain all the clustering information. Second, we use heuristics from these analyses to design the Data Spectroscopic clustering (DaSpec) algorithm that uses properly selected top eigenvectors, determines the number of clusters, gives data labels, and provides a classification rule for future data, all based on only one eigen decomposition. Our findings not only extend and go beyond the intuitions underlying existing spectral techniques (e.g. spectral clustering and Kernel Principal Components Analysis), but also provide insights about their usability and modes of failure. Simulation studies and experiments on real world data are conducted to show the promise of our proposed data spectroscopy clustering algorithm relative to k-means and one spectral method. In particular, DaSpec seems to be able to handle unbalanced groups and recover clusters of different shapes better than competing methods.