On the Probabilistic Approximation in Reproducing Kernel Hilbert Spaces

TL;DR

This paper explores probabilistic function approximation in reproducing kernel Hilbert spaces, establishing theoretical foundations, generalizing the representer theorem, and linking the problem to measure quantization and sampling theory.

Contribution

It introduces existence and uniqueness results, extends the representer theorem to probabilistic settings, and connects approximation to measure quantization in finite and infinite-dimensional spaces.

Findings

Existence and uniqueness of the optimizer under mild assumptions.

Generalization of the representer theorem to probabilistic approximation.

Connection between probabilistic approximation and measure quantization in finite-dimensional cases.

Abstract

This paper studies the probabilistic function approximation problem over reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer under mild assumptions. Furthermore, we generalize the celebrated representer theorem to our setting, and especially when the probability measure is finitely supported, or the Hilbert space is finite-dimensional, we show that the probabilistic approximation problem turns out to be a measure quantization problem, which connects the probabilistic function approximation to the sampling theory. Some discussions and examples are also given when the reproducing kernel Hilbert space is infinite-dimensional and the measure is infinitely supported.