Robust dimension-free Gram operator estimates

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.