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February 2021 Hanson–Wright inequality in Hilbert spaces with application to $K$-means clustering for non-Euclidean data
Xiaohui Chen, Yun Yang
Bernoulli 27(1): 586-614 (February 2021). DOI: 10.3150/20-BEJ1251

Abstract

We derive a dimension-free Hanson–Wright inequality for quadratic forms of independent sub-gaussian random variables in a separable Hilbert space. Our inequality is an infinite-dimensional generalization of the classical Hanson–Wright inequality for finite-dimensional Euclidean random vectors. We illustrate an application to the generalized $K$-means clustering problem for non-Euclidean data. Specifically, we establish the exponential rate of convergence for a semidefinite relaxation of the generalized $K$-means, which together with a simple rounding algorithm imply the exact recovery of the true clustering structure.

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Xiaohui Chen. Yun Yang. "Hanson–Wright inequality in Hilbert spaces with application to $K$-means clustering for non-Euclidean data." Bernoulli 27 (1) 586 - 614, February 2021. https://doi.org/10.3150/20-BEJ1251

Information

Received: 1 January 2020; Revised: 1 July 2020; Published: February 2021
First available in Project Euclid: 20 November 2020

zbMATH: 07282863
MathSciNet: MR4177382
Digital Object Identifier: 10.3150/20-BEJ1251

Keywords: $k$-means , Hanson–Wright inequality , Hilbert space , semidefinite relaxation

Rights: Copyright © 2021 Bernoulli Society for Mathematical Statistics and Probability

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Vol.27 • No. 1 • February 2021
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