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.
Citation
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
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