Speaker
Description
Due to the size of many kernel matrices that arise in applications, it is often necessary to work with their low-rank approximations in order to efficiently perform many computations. Low-rank matrix decompositions to such matrices may be quickly obtained by exploiting the analytic structure of the underlying kernel, for example by using Taylor expansions or an integral representation; such ideas trace back to the fast multipole method. However, there is often a gap between the theoretical epsilon-rank of a given kernel matrix, explicitly obtained using its eigenvalue decomposition, and the existing analytic decompositions of the same rank. In this work, we aim to bridge this gap by exploring the proxy point method, which approximates complex-analytic kernel matrices using a discretized contour integral representation. In particular, we combine this approach with a judicious choice of conformal map applied to the set of points at which the kernel is evaluated in order to obtain effective analytic approximations. We apply this new method to some well-studied kernel matrices, such as the Hilbert and Cauchy matrices, in order to compare it with existing analytic methods and with the best available theoretical bounds. In the process, we demonstrate a novel way of obtaining one kind of theoretically-optimal solution to a related problem in rational approximation.