An analysis of Ermakov-Zolotukhin quadrature using kernels

TL;DR

This paper analyzes Ermakov-Zolotukhin quadrature using kernel methods, revealing its relation to optimal kernel quadrature and providing new theoretical error bounds based on eigenvalues.

Contribution

It establishes a connection between Ermakov-Zolotukhin quadrature and optimal kernel quadrature, deriving a tractable error formula involving kernel eigenvalues.

Findings

Derived a formula for expected squared worst-case error involving kernel eigenvalues.

Showed how Ermakov-Zolotukhin quadrature relates to determinantal point processes.

Improved theoretical guarantees for kernel quadrature methods.

Abstract

We study a quadrature, proposed by Ermakov and Zolotukhin in the sixties, through the lens of kernel methods. The nodes of this quadrature rule follow the distribution of a determinantal point process, while the weights are defined through a linear system, similarly to the optimal kernel quadrature. In this work, we show how these two classes of quadrature are related, and we prove a tractable formula of the expected value of the squared worst-case integration error on the unit ball of an RKHS of the former quadrature. In particular, this formula involves the eigenvalues of the corresponding kernel and leads to improving on the existing theoretical guarantees of the optimal kernel quadrature with determinantal point processes.