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

Understand

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.