Data spectroscopy: Eigenspaces of convolution operators and clustering

Details Data spectroscopy: Eigenspaces of convolution operators and clustering Data spectroscopy: Eigenspaces of convolution operators and clustering

2008-07-23

arxiv.org/abs/0807.3719v2

Annals of Statistics 2009, Vol. 37, No. 6B, 3960-3984

Statistics

Machine Learning

Published in at http://dx.doi.org/10.1214/09-AOS700 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/09-AOS700

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.

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