General reproducing properties in RKHS with application to derivative and integral operators

TL;DR

This work develops a unified framework for reproducing properties in RKHS that extends beyond the classic evaluation functional to broader linear operators, including derivatives and integrals. It introduces the closure of finite combinations of composition operators and provides a necessary-and-sufficient criterion, via the Loève criterion, for the generalized reproducing property to hold and for operator representers to converge. The derivative reproducing property is established under mild kernel regularity, namely $K$ being $C^1$ with a continuous cross-derivative near the diagonal, yielding $f'(x) = \langle f, \partial K/\partial x_1(x, \cdot)\rangle$ and a corresponding norm identity, under which all functions in the RKHS are differentiable. Additionally, the paper proves a minimal condition for mean embedding in RKHS with improper Riemann integrals: if $\iint |K(x,y)| p(x)p(y) \, dx\, dy$ is finite, then $\mu_p(x) = \int K(x,y) p(y) \, dy$ exists in $\mathcal{H}$ and satisfies the RKHS moment relation $\mathbb{E}_{X\sim p}[f(X)] = \langle f, \mu_p \rangle$, with a weaker requirement than the usual Lebesgue-style criterion and a concrete kernel example illustrating the relaxation. These results enable representer-theorem-style solutions for learning with function values, gradients, or other operators within a broad operator class.

Abstract

In this paper, we consider the reproducing property in Reproducing Kernel Hilbert Spaces (RKHS). We establish a reproducing property for the closure of the class of combinations of composition operators under minimal conditions. This allows to revisit the sufficient conditions for the reproducing property to hold for the derivative operator, as well as for the existence of the mean embedding function. These results provide a framework of application of the representer theorem for regularized learning algorithms that involve data for function values, gradients, or any other operator from the considered class.

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Proposition 1: Loève criterion
• proof
• Theorem 1
• proof
• Theorem 2
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thm:derivative']}