Robust dimension-free Gram operator estimates

TL;DR

This paper develops a robust, dimension-free estimator for the Gram operator in Hilbert spaces, providing theoretical guarantees and demonstrating improved performance over classical methods with heavy-tailed data.

Contribution

It introduces a novel robust estimation method for the Gram operator that is dimension-free and applicable in infinite-dimensional Hilbert spaces, with theoretical bounds and empirical validation.

Findings

The estimator achieves uniform bounds under weak moment assumptions.

It outperforms classical empirical estimators with heavy-tailed data.

The approach extends finite-dimensional bounds to infinite-dimensional settings.

Abstract

In this paper we investigate the question of estimating the Gram operator by a robust estimator from an i.i.d. sample in a separable Hilbert space and we present uniform bounds that hold under weak moment assumptions. The approach consists in first obtaining non-asymptotic dimension-free bounds in finite-dimensional spaces using some PAC-Bayesian inequalities related to Gaussian perturbations of the parameter and then in generalizing the results in a separable Hilbert space. We show both from a theoretical point of view and with the help of some simulations that such a robust estimator improves the behavior of the classical empirical one in the case of heavy tail data distributions.