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Spectral Properties of Radial Kernels and Clustering in High Dimensions

David Cohen-Steiner 1 Alba Chiara de Vitis 1 
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : In this paper, we study the spectrum and the eigenvectors of radial kernels for mixtures of distributions in R n. Our approach focuses on high dimensions and relies solely on the concentration properties of the components in the mixture. We give several results describing of the structure of kernel matrices for a sample drawn from such a mixture. Based on these results, we analyze the ability of kernel PCA to cluster high dimensional mixtures. In particular, we exhibit a specific kernel leading to a simple spectral algorithm for clustering mixtures with possibly common means but different covariance matrices. We show that the minimum angular separation between the covariance matrices that is required for the algorithm to succeed tends to 0 as n goes to infinity.
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Submitted on : Monday, January 6, 2020 - 5:27:15 PM
Last modification on : Friday, February 4, 2022 - 3:21:53 AM
Long-term archiving on: : Tuesday, April 7, 2020 - 10:33:40 PM


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  • HAL Id : hal-01969956, version 7


David Cohen-Steiner, Alba Chiara de Vitis. Spectral Properties of Radial Kernels and Clustering in High Dimensions. {date}. ⟨hal-01969956v7⟩



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