Learning convolution operators on compact Abelian groups

2025-01-09

AI Summary

This paper addresses the problem of learning convolutional kernels on compact abelian groups. To estimate convolution operators within translation-invariant reproducing kernel Hilbert spaces (RKHS), we propose a regularization-based learning method rooted in ridge regression. Our key conceptual contribution is to reinterpret the classical ridge regression regularity assumption as spatial-frequency localization on the group domain—a novel perspective that establishes, for the first time, a systematic bridge between harmonic analysis and statistical learning theory. Theoretically, we derive an explicit finite-sample upper bound on the estimation error of the convolution kernel. Numerical experiments confirm that this bound accurately captures the empirical convergence behavior. Overall, our work provides a new framework for operator learning on groups that is both theoretically rigorous and computationally tractable.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

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, discussing natural regularity condition 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.

Problem

• Mathematical Groups
• Convolution Operation
• Learning Algorithms

Innovation

• Convolution Learning
• Regularization in Ridge Regression
• Translation-Invariant Hilbert Spaces