Covariance discriminative power of kernel clustering methods

Abla Kammoun, Romain Couillet

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish (US)
JournalElectronic Journal of Statistics
Issue number1
StatePublished - Jan 26 2023

Bibliographical note

KAUST Repository Item: Exported on 2023-02-02
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.

ASJC Scopus subject areas

  • Statistics and Probability


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