Intrinsic-dimension empirical Bernstein inequalities for bounded self-adjoint operators
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, self-adjoint Hilbert-Schmidt 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 with a sharper first-order asymptotic radius for non-isotropic random matrices and seamlessly extends 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.
Related
- Empirical Bernstein in smooth Banach spaces
- Bernstein’s inequalities for general Markov chains
- Weighted Bernstein-type inequalities on a trapezoidal domain
- Jackson and Bernstein Inequalities with Explicit Weight Dependence for Band-Limited Wavelet Approximation in Weighted Besov Spaces
- Sharp Bernstein--Nikolskii inequalities for polynomials and entire functions of exponential type
BibTeX
@misc{martineztaboada2026intrinsicdimension,
title = {{Intrinsic-dimension empirical Bernstein inequalities for bounded self-adjoint operators}},
author = {Diego Martinez-Taboada and Aaditya Ramdas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.15278},
eprint = {2605.15278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.15278},
}
RIS
TY - UNPB
TI - Intrinsic-dimension empirical Bernstein inequalities for bounded self-adjoint operators
AU - Diego Martinez-Taboada
AU - Aaditya Ramdas
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2605.15278
UR - https://arxiv.org/abs/2605.15278
AB - 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, self-adjoint Hilbert-Schmidt 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 with a sharper first-order asymptotic radius for non-isotropic random matrices and seamlessly extends 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.
ER -