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.

翻译：我们从核方法的视角研究埃马科夫与佐洛图欣于二十世纪六十年代提出的一种求积方法。该求积规则的节点服从行列式点过程的分布，而权值通过线性系统定义，与最优核求积法类似。本研究揭示了这两类求积法之间的关联，并推导出前一种求积法在单位球上平方最坏情形积分误差期望值的可计算公式，该公式涉及所对应核的特征值，进而改进了现有基于行列式点过程的最优核求积法的理论保证。