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

Xiaohui Chen, Yun Yang|arXiv (Cornell University)|Oct 26, 2018

Sparse and Compressive Sensing Techniques28 references4 citations

TL;DR

This paper establishes a dimension-free Hanson-Wright inequality for quadratic forms of sub-Gaussian random variables in separable Hilbert spaces, generalizing the classical finite-dimensional result. It applies this inequality to prove exponential convergence rates for a semidefinite relaxation of generalized K-means clustering on non-Euclidean data, enabling exact recovery via a rounding algorithm.

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.

Motivation & Objective

• To derive a dimension-free Hanson-Wright inequality for sub-Gaussian random variables in separable Hilbert spaces, extending classical finite-dimensional results.
• To address the challenge of clustering non-Euclidean data with non-linear features using computationally tractable, polynomial-time algorithms.
• To establish strong statistical guarantees—specifically, exponential rate of convergence—for a semidefinite relaxation of the generalized K-means clustering problem.
• To demonstrate that the semidefinite relaxation, combined with a simple rounding procedure, achieves exact recovery of the true clustering structure.
• To provide a principled alternative to heuristic greedy algorithms commonly used in kernel clustering with theoretical recovery guarantees.

Proposed method

• Derives a Hilbert space version of the Hanson-Wright inequality for centered sub-Gaussian random variables taking values in a separable Hilbert space, with norms defined via trace and operator norms.
• Introduces the concept of sub-Gaussian random variables in Hilbert spaces using a trace-class covariance operator Γ, defining the sub-Gaussian norm via the operator norm and trace norm.
• Applies the infinite-dimensional Hanson-Wright inequality to analyze the concentration of quadratic forms in the context of generalized K-means clustering with non-Euclidean data.
• Uses the inequality to establish an exponential tail bound on the deviation of the semidefinite relaxation objective from its expectation.
• Employs a monotone rearrangement lemma (Lemma A.5) to control the error in the clustering objective and derive convergence rates.
• Combines the concentration bound with a rounding algorithm to show that the optimal solution of the SDP relaxation recovers the true clustering structure exactly with high probability.

Experimental results

Research questions

• RQ1 Can the classical Hanson-Wright inequality be extended to infinite-dimensional Hilbert spaces for sub-Gaussian random variables?
• RQ2 Does the resulting infinite-dimensional Hanson-Wright inequality yield dimension-free concentration bounds for quadratic forms in Hilbert spaces?
• RQ3 Can this generalized inequality be used to establish strong statistical guarantees for semidefinite relaxation in generalized K-means clustering on non-Euclidean data?
• RQ4 Does the semidefinite relaxation of the generalized K-means problem achieve exact recovery of the true clustering structure under sub-Gaussian noise?
• RQ5 Can a simple rounding algorithm recover the exact clustering from the SDP solution with high probability, given the concentration bounds?

Key findings

• A dimension-free Hanson-Wright inequality is established for quadratic forms of sub-Gaussian random variables in a separable Hilbert space, with tail bounds depending only on the sub-Gaussian norm and operator norms of the matrix.
• The inequality is proven via a truncation argument and convergence of finite-dimensional approximations, relying on the monotone convergence theorem and properties of trace-class operators.
• The exponential tail bound for the quadratic form is of the form $ ext{Pr}(|X^T A X - ext{E}[X^T A X]| angle t) angle angle 2 ext{exp}ig[-C ext{min}(t^2 / (L^4 Vert A Vert_{ ext{HS}}^2), t / (L^2 Vert A Vert_{ ext{op}}))ig] $, where $ L $ is the sub-Gaussian norm.
• For the generalized K-means clustering problem in Hilbert spaces, the semidefinite relaxation achieves an exponential rate of convergence in the deviation of the objective function from its expectation.
• The concentration bound implies that the optimal solution of the SDP relaxation recovers the true clustering structure exactly with high probability when the noise level is sufficiently low.
• A simple rounding algorithm applied to the SDP solution achieves exact recovery of the true clustering, providing a computationally tractable and statistically consistent method for non-Euclidean data clustering.