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Learning convolution operators on compact Abelian groups

Emilia Magnani, Ernesto De Vito, Philipp Hennig, Lorenzo Rosasco

TL;DR

The paper develops a regularized framework for learning convolution operators defined by kernels on compact Abelian groups, formulating the problem as ridge regression in translation-invariant Hilbert spaces. By leveraging Fourier analysis, it derives explicit per-frequency estimators and non-asymptotic error bounds that tie kernel regularity to space/frequency localization and the spectral decay of input signals. The main theoretical contribution is a set of rates for the estimator in terms of a source condition parameter $r$ and a decay parameter $b$, with high-probability bounds for both the kernel-norm error and the operator-prediction error. Numerical experiments validate the theory and reveal how input localization influences recovery quality and the practical viability of heat-kernel approximation for PDEs. The work highlights the interplay between localization properties and learning performance, offering a principled approach to learning Green functions and impulse responses in structured, group-based settings.

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.

Learning convolution operators on compact Abelian groups

TL;DR

The paper develops a regularized framework for learning convolution operators defined by kernels on compact Abelian groups, formulating the problem as ridge regression in translation-invariant Hilbert spaces. By leveraging Fourier analysis, it derives explicit per-frequency estimators and non-asymptotic error bounds that tie kernel regularity to space/frequency localization and the spectral decay of input signals. The main theoretical contribution is a set of rates for the estimator in terms of a source condition parameter and a decay parameter , with high-probability bounds for both the kernel-norm error and the operator-prediction error. Numerical experiments validate the theory and reveal how input localization influences recovery quality and the practical viability of heat-kernel approximation for PDEs. The work highlights the interplay between localization properties and learning performance, offering a principled approach to learning Green functions and impulse responses in structured, group-based settings.

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.
Paper Structure (27 sections, 14 theorems, 165 equations, 4 figures)

This paper contains 27 sections, 14 theorems, 165 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. Background on group theory and Harmonic Analysis
  5. Learning convolution operators with regularization
  6. Statistical model
  7. Translation invariant Hilbert spaces
  8. Ridge regression
  9. Theoretical analysis of the learning error
  10. Main results
  11. A priori assumptions and space/frequency localization
  12. Discussion
  13. Numerical simulations
  14. Error behavior and localization
  15. Setup
  16. ...and 12 more sections

Key Result

Theorem 4.1

Assume that the positive part of the spectrum of $\Sigma$ is denumerable, i.e. for some injective map $I\ni \ell \mapsto \xi_\ell \in \widehat{G}$. Moreover, suppose that, for some $0\leq r\leq 1/2$, $w_*\in\mathcal{H}$ satisfies the source condition and, for some $b\in [1,+\infty]$, the family $(\sigma_{\xi_\ell})_{\ell\in I}$ satisfies the decay condition Set where $\kappa=D_XD_K$. For any

Figures (4)

  • Figure 1: Error decay. (Left) $\|w_n^\lambda-w_*\|^2_\mathcal{H}$. (Right) $\|\Sigma^{\frac{1}{2}}(w_n^\lambda-w_*)\|_\mathcal{H}^2$. Each curve compares frequency-localized vs. space-localized inputs. Dotted lines indicate the theoretical convergence rates for reference.
  • Figure 2: Example of reconstruction. Comparison of the true $w_\ast$ with the estimated $w_n^{\lambda_n}$ for $n=50$ in both input scenarios.
  • Figure 3: Heat kernel reconstruction after $n=15$ input samples for different values od $\delta$.
  • Figure 4: Corresponding convolution operators (circulant matrices).

Theorems & Definitions (41)

  • Example 1: Torus
  • Example 2: Circulant matrices
  • Example 3: Frequency localization
  • Example 4: space localization
  • Remark 1
  • Remark 2
  • Example 5: Periodic Sobolev spaces
  • Example 6: Exponential decay on the torus
  • Example 7: Trigonometric polynomials
  • Remark 3
  • ...and 31 more