RKHS Representation of Algebraic Convolutional Filters with Integral Operators

TL;DR

A comprehensive theory is developed 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, providing a principled foundation for learnable filters in integral-operator-based neural architectures.

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.

Figures (5)

• Figure 1: Illustration for Example \ref{['exm_cor_gphon_fourier_kv']}. Left: Discrete representation showing eigenfunctions $\varphi_{1}(u)$, $\varphi_{4}(u)$, and $\varphi_{7}(u)$ (curves) evaluated at sample points $u \in \{0.2, 0.45, 0.7, 0.86\}$ (markers), with coefficient vector $\boldsymbol{\alpha}=[-2,1,-0.5,0.2]$ indicated by $\alpha_v$. Center: Graphon signal $f(u)$ and kernel functions $k_{v}(u)$ for $v \in \{0.2, 0.45, 0.7, 0.86\}$ as functions of $u \in [0,1]$. Right: Normalized Fourier coefficients $\widehat{f}_{i}/\lambda_{i}^{2}$.
• Figure 2: Pictorial representation of a generic algebraic signal model (ASM) $(\ccalA, \ccalM, \rho)$. Filters are elements of the algebra $\ccalA$ while the signals are the elements of the vector space $\ccalH$. The homomorphism, $\rho$, translates the abstract filters in $\ccalA$ into concrete operators in $\text{End}(\ccalH)$, that act on the signals in $\ccalH$. The symbol $\text{End}(\ccalH)$ represents the space of linear operators from $\ccalH$ onto itself.
• Figure 3: Illustration diagram of the results in Theorem \ref{['thm_TKn_vs_Kboxn']} and Corollary \ref{['cor_diff_as_sum_Hkn']}. The operator $\boldsymbol{T}^{r}_{K}$ maps the signal $f\in\ccalH(K)$ on the RKHS $\ccalH\left(K^{\square (r+1)}\right)$, where the resultant diffused signal can be written as an expansion in terms of the kernel functions $k_{\ell_{r}}^{(r+1)}$. Then, the result of diffusing $f$ with $p(\boldsymbol{T}_{K})=\sum_{r=0}^{R}h_{r}\boldsymbol{T}_{K}^{r}f$ is obtained as a weighted sum of signals on the RKHS spaces $\{ \ccalH(K^{\square (r+1)}) \}_{r=0}^{R}$, where the weights are the filter coefficients $\{ h_{r} \}_{r=0}^{R}$.
• Figure 4: Diagramatic illustration of the consequences of Corollary \ref{['cor_rkhs_uncertainty']}. An RKHS with a finite expansion in terms of the $k_{t}$ functions cannot be exactly bandlimited. However, due to the nature of the eigenvalues, $\sigma_i$, of the operator $\boldsymbol{T}_{K}$ the signal can be approximately bandlimited. Given the decomposition of a signal $f\in\ccalH(K)$ with the representation in \ref{['eq_f_decomp_mult_bwd']}, the tradeoff between RKHS finiteness and bandlimited frequencies is determined by $\vert\ccalT\vert-B$. If $B<\vert\ccalT\vert$ it is possible to choose $\vert\ccalT\vert -B$ coefficients $a_{t}$ to minimize the size of $f_{B+1 : \vert\ccalT\vert}$ in \ref{['eq_f_decomp_mult_bwd']}. Then, $B$ coefficients determine a specific low pass behavior, while the remaining coefficients are selected to reduce the size of the residuals (green shaded area on the right side). This implies that, the larger $\vert\ccalT\vert - B$ is, the faster the residuals can go to zero.
• Figure 5: Illustration associated to Example \ref{['exmpl_rkhs_reg_filter']}. Using the representer theorem we find the optimal polynomial filter $p(t)$, such that $p(\sigma_{i})=\exp(-(\sigma_{i}-\sigma_{c})^2/\gamma)$ where $\sigma_c$ and $\gamma$ are fixed parameters. From Corollary \ref{['cor_filt_representer']}, the optimal filter can be written as $p^{\ast}(u)=\sum_{i=1}^{q}a_{i}k_{\sigma_{i}}(u)$. The pictures show the resultant filter, when considering multiple values of $q$ -- the number of points used in the regression -- and multiple values of $\sigma_{c}$ (the localization of the maximum point of the filters).

Theorems & Definitions (24)

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