On the Probabilistic Approximation in Reproducing Kernel Hilbert Spaces

This paper generalizes the least square method to probabilistic approximation in reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer. Furthermore, we generalize the celebrated representer theorem in this setting, and especially when the probability measure is fin...

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Bibliographic Details
Published in:arXiv.org 2024-09
Main Authors: Chen, Dongwei, Wang, Kai-Hsiang
Format: Article
Language:English
Subjects:
Approximation
Hilbert space
Probability theory
Statistical analysis
Uniqueness theorems
Online Access:Get full text
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Summary:This paper generalizes the least square method to probabilistic approximation in reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer. Furthermore, we generalize the celebrated representer theorem in this setting, and especially when the probability measure is finitely supported, or the Hilbert space is finite-dimensional, we show that the approximation problem turns out to be a measure quantization problem. Some discussions and examples are also given when the space is infinite-dimensional and the measure is infinitely supported.
ISSN:2331-8422