Abstract: We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d. samples from the underlying measure and evaluation of the kernel and that result in a small worst-case error. In addition to our theoretical results and the benefits resulting from convex weights, our experiments indicate that this construction can compete with the optimal bounds in well-known examples.
Furthermore, we will talk about a refined error analysis of Nyström approximation and a consequent theoretical guarantee of the proposed kernel quadrature, which improves the results already published in NeurIPS 2022. This is joint work with Harald Oberhauser and Terry Lyons.