Estimation of linear operators from scattered impulse responses

TL;DR

This paper introduces a novel regularized estimator for integral operators with smooth kernels, using reproducing kernel Hilbert spaces, that is robust to noise and effective in high-dimensional settings.

Contribution

It presents a new, numerically feasible estimator for integral operators from scattered data, with proven minimax optimality and robustness to noise.

Findings

Estimator is robust to noise.

Achieves minimax optimality.

Effective in large-dimensional problems.

Abstract

We provide a new estimator of integral operators with smooth kernels, obtained from a set of scattered and noisy impulse responses. The proposed approach relies on the formalism of smoothing in reproducing kernel Hilbert spaces and on the choice of an appropriate regularization term that takes the smoothness of the operator into account. It is numerically tractable in very large dimensions. We study the estimator's robustness to noise and analyze its approximation properties with respect to the size and the geometry of the dataset. In addition, we show minimax optimality of the proposed estimator.