Covariance discriminative power of kernel clustering methods

Abstract
Let x1, ··· , xn be independent observations of size p, each of them belonging to one of c distinct classes. We assume that observations within the class a are characterized by their distribution N (0, 1 pCa) where here C1, ··· , Cc are some non-negative definite p × p matrices. This paper studies the asymptotic behavior of the symmetric matrix Φ˜kl = √p (xT k xl)2δk=l when p and n grow to infinity with n p → c0. Particularly, we prove that, if the class covariance matrices are sufficiently close in a certain sense, the matrix Φ behaves like a low-rank perturbation of a ˜ Wigner matrix, presenting possibly some isolated eigenvalues outside the bulk of the semi-circular law. We carry out a careful analysis of some of the isolated eigenvalues of Φ and their associated eigenvectors and illustrate ˜ how these results can help understand spectral clustering methods that use Φ as a kernel matrix.

Citation
Kammoun, A., & Couillet, R. (2023). Covariance discriminative power of kernel clustering methods. Electronic Journal of Statistics, 17(1). https://doi.org/10.1214/23-ejs2107

Acknowledgements
The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST). The work of Couillet is supported by the ANR Project RMT4GRAPH (ANR-14-CE28-0006) and the HUAWEI RMTin5G project.The authors would like to deeply thank an anonymous reviewer for his careful reading and valuable comments, which helped us to improve the quality of the manuscript.

Publisher
Institute of Mathematical Statistics

Journal
Electronic Journal of Statistics

DOI
10.1214/23-ejs2107

Additional Links
https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-17/issue-1/Covariance-discriminative-power-of-kernel-clustering-methods/10.1214/23-EJS2107.full

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