Learning convolution operators on compact Abelian groups

TL;DR

This paper investigates learning convolution operators on compact Abelian groups using regularization, providing finite sample guarantees and linking classical regularity assumptions to space/frequency localization.

Contribution

It introduces a regularization-based method for learning convolution operators on compact Abelian groups with theoretical guarantees and novel interpretation of regularity conditions.

Findings

Finite sample bounds for the ridge regression estimator.

Regularity assumptions relate to space/frequency localization.

Numerical simulations validate theoretical results.

Abstract

We consider the problem of learning convolution operators associated to compact Abelian groups. We study a regularization-based approach and provide corresponding learning guarantees under natural regularity conditions on the convolution kernel. More precisely, we assume the convolution kernel is a function in a translation invariant Hilbert space and analyze a natural ridge regression (RR) estimator. Building on existing results for RR, we characterize the accuracy of the estimator in terms of finite sample bounds. Interestingly, regularity assumptions which are classical in the analysis of RR, have a novel and natural interpretation in terms of space/frequency localization. Theoretical results are illustrated by numerical simulations.