On the Probabilistic Approximation in Reproducing Kernel Hilbert Spaces

2024/09/18 by Dongwei Chen, Chen, Dongwei, Kai-Hsiang Wang +1
Computer Science · #46E22 #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods

paper · pdf · doi:10.48550/arxiv.2409.11679

openalex publication_date 2024/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Scalable Learning from Probability Measures with Mean Measure Quantization
• Random Bit Quadrature and Approximation of Distributions on Hilbert\n Spaces
• Sample-based almost-sure quasi-optimal approximation in reproducing kernel Hilbert spaces
• Stochastic subspace correction in Hilbert space
• n-Best Kernel Approximation in Reproducing Kernel Hilbert Spaces