Hanson–Wright inequality in Hilbert spaces with application to $K$-means clustering for non-Euclidean data

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