RKHS Representation of Algebraic Convolutional Filters with Integral Operators

2026-02-22

AI Summary

This work addresses the lack of a systematic connection between classical integral operators and reproducing kernel Hilbert spaces (RKHS) in algebraic signal processing, which has hindered theoretical advances in graph signal processing and learnable filters. The paper establishes, for the first time, an algebraic correspondence between integral operators and RKHS by constructing a unital kernel algebra via the box product of operator symbols, thereby deriving the associated reproducing kernel and characterizing its spectral and algebraic properties. This framework enables exact alignment between graph signal spectral decomposition and RKHS representation, extends naturally to directed graphs, and proves that when the spectral domain of a regularized learning problem is a subset of the signal domain, the optimal filter admits a finite-dimensional RKHS representation—providing a rigorous theoretical foundation for learnable filters in neural architectures based on integral operators.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Graphical Models
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Integral operators play a central role in signal processing, underpinning classical convolution, and filtering on continuous network models such as graphons. While these operators are traditionally analyzed through spectral decompositions, their connection to reproducing kernel Hilbert spaces (RKHS) has not been systematically explored within the algebraic signal processing framework. In this paper, we develop a comprehensive theory showing that the range of integral operators naturally induces RKHS convolutional signal models whose reproducing kernels are determined by a box product of the operator symbols. We characterize the algebraic and spectral properties of these induced RKHS and show that polynomial filtering with integral operators corresponds to iterated box products, giving rise to a unital kernel algebra. This perspective yields pointwise RKHS representations of filters via the reproducing property, providing an alternative to operator-based implementations. Our results establish precise connections between eigendecompositions and RKHS representations in graphon signal processing, extend naturally to directed graphons, and enable novel spatial--spectral localization results. Furthermore, we show that when the spectral domain is a subset of the original domain of the signals, optimal filters for regularized learning problems admit finite-dimensional RKHS representations, providing a principled foundation for learnable filters in integral-operator-based neural architectures.

Problem

• integral operators
• reproducing kernel Hilbert spaces
• algebraic signal processing
• convolutional filters
• graphons

Innovation

• RKHS
• integral operators
• box product
• graphon signal processing
• learnable filters