Positively Weighted Kernel Quadrature via Subsampling
This work addresses a specific problem in numerical integration for researchers in computational mathematics, offering incremental improvements in algorithm design.
The paper tackles the problem of constructing kernel quadrature rules with convex weights using only i.i.d. samples and kernel evaluations, resulting in algorithms that achieve small worst-case error and competitive performance with optimal bounds in experiments.
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.