Intrinsic-dimension empirical Bernstein inequalities for bounded self-adjoint operators

TL;DR

This paper introduces data-driven, intrinsic-dimension-based Bernstein inequalities for sums of bounded self-adjoint operators, improving high-dimensional operator concentration bounds.

Contribution

It establishes the first empirical Bernstein inequalities that replace unknown variance with empirical estimates, relying on intrinsic rather than ambient dimension.

Findings

Achieves asymptotic sharpness with oracle rates

Provides dimension-free, computable concentration bounds

Extends to infinite-dimensional Hilbert spaces

Abstract

Operator-valued concentration inequalities are foundational to the analysis of modern high-dimensional statistics and randomized algorithms. However, standard oracle bounds are frequently limited in practice: they require explicit a priori knowledge of the true variance, and often explicitly scale with the ambient dimension, rendering them vacuous for infinite-dimensional or heavily structured operators. Motivated by these challenges, we establish the first empirical Bennett and Bernstein inequalities for sums of independent, bounded, compact self-adjoint operators. Our fully data-driven bounds replace the unknown variance with an empirical estimate and rely strictly on the intrinsic dimension rather than the ambient dimension. This structural shift yields computable, dimension-free guarantees that are strictly sharper for non-isotropic random matrices and seamlessly extend to infinite-dimensional Hilbert spaces. We demonstrate that our empirical bounds achieve asymptotic sharpness with the best known oracle rates. Finally, as an independent byproduct, we derive novel empirical concentration guarantees for the intrinsic dimension itself.