On the Probabilistic Approximation in Reproducing Kernel Hilbert Spaces

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.