# RKHS Representation of Algebraic Convolutional Filters with Integral Operators

arXiv: https://arxiv.org/abs/2602.19094 · HTML: https://mlsanity.com/paper/2602.19094/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 32 references that are in the map.
- First posted on arXiv 22 Feb 2026.

## Builds on (most cited first, 18 of 18)

- [Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nystr"om Method, and Use of Kernels in Machine Learning: Tutorial and Survey](https://mlsanity.com/paper/2106.08443/) (2021)
- [Algebraic Neural Networks: Stability to Deformations](https://mlsanity.com/paper/2009.01433/) (2020)
- [Graphon Filters: Graph Signal Processing in the Limit](https://mlsanity.com/paper/2003.02099/) (2020)
- [Causal Fourier Analysis on Directed Acyclic Graphs and Posets](https://mlsanity.com/paper/2209.07970/) (2022)
- [Graphon Pooling in Graph Neural Networks](https://mlsanity.com/paper/2003.01795/) (2020)
- [Convolutional Learning on Multigraphs](https://mlsanity.com/paper/2209.11354/) (2022)
- [Discrete Signal Processing with Set Functions](https://mlsanity.com/paper/2001.10290/) (2020)
- [Stability of Algebraic Neural Networks to Small Perturbations](https://mlsanity.com/paper/2010.11544/) (2020)
- [Discrete Signal Processing on Meet/Join Lattices](https://mlsanity.com/paper/2012.04358/) (2020)
- [Convolutional Filtering and Neural Networks with Non Commutative Algebras](https://mlsanity.com/paper/2108.09923/) (2021)
- [Graphon Pooling for Reducing Dimensionality of Signals and Convolutional Operators on Graphs](https://mlsanity.com/paper/2212.08171/) (2022)
- [Lie Group Algebra Convolutional Filters](https://mlsanity.com/paper/2305.04431/) (2023)
- [Learning with Multigraph Convolutional Filters](https://mlsanity.com/paper/2210.16272/) (2022)
- [Sampling and Uniqueness Sets in Graphon Signal Processing](https://mlsanity.com/paper/2401.06279/) (2024)
- [Algebraic Convolutional Filters on Lie Group Algebras](https://mlsanity.com/paper/2210.17425/) (2022)
- [Mean Field Games on Weighted and Directed Graphs via Colored Digraphons](https://mlsanity.com/paper/2209.03887/) (2022)
- [Convolutional Filtering with RKHS Algebras](https://mlsanity.com/paper/2411.01341/) (2024)
- [Efficient Evolutionary Models with Digraphons](https://mlsanity.com/paper/2104.12748/) (2021)

## Built on by (most cited first, 1 of 1)

- [Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensions](https://mlsanity.com/paper/2609.36106/) (2026)

## Abstract (arXiv)

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.