Learning convolution operators on compact Abelian groups
This work addresses a theoretical problem in machine learning for researchers, but it appears incremental as it builds on existing ridge regression results.
The paper tackles the problem of learning convolution operators on compact Abelian groups using a ridge regression estimator, providing finite sample bounds under regularity conditions on the kernel, with results illustrated by numerical simulations.
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.