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 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.