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[2602.19094] RKHS Representation of Algebraic Convolutional Filters with Integral Operators
Summary
This paper explores the RKHS representation of algebraic convolutional filters using integral operators, establishing connections between spectral decompositions and convolutional signal models.
Why It Matters
Understanding the RKHS representation of algebraic convolutional filters is crucial for advancing signal processing techniques, particularly in machine learning applications. This research provides a theoretical foundation that can enhance the design of learnable filters in neural architectures, impacting various fields including graph signal processing and data analysis.
Key Takeaways
- Integral operators are key in signal processing and filtering.
- The paper establishes a theory linking RKHS convolutional models to integral operators.
- Polynomial filtering corresponds to iterated box products in this framework.
- The findings extend to directed graphons, enhancing spatial-spectral localization.
- Optimal filters for learning problems can be represented in finite-dimensional RKHS.
Computer Science > Machine Learning arXiv:2602.19094 (cs) [Submitted on 22 Feb 2026] Title:RKHS Representation of Algebraic Convolutional Filters with Integral Operators Authors:Alejandro Parada-Mayorga, Alejandro Ribeiro, Juan Bazerque View a PDF of the paper titled RKHS Representation of Algebraic Convolutional Filters with Integral Operators, by Alejandro Parada-Mayorga and 2 other authors View PDF 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 ...
