Minimax separation of the Cauchy kernel

TL;DR

This paper develops an optimal low-rank approximation method for the Cauchy kernel over separated real domains, including algorithms and heuristics, with applications to numerical stability and efficiency.

Contribution

It introduces a new skeleton decomposition approach for the Cauchy kernel, optimizing parameters and analyzing stability, advancing kernel approximation techniques.

Findings

Optimal low-rank approximation achieved

Effective heuristic algorithms developed

Numerical stability forms identified

Abstract

We prove and apply an optimal low-rank approximation of the Cauchy kernel over separated real domains. A skeleton decomposition is the minimum over real-valued functions of the maximum relative pointwise error. We present an algorithm to optimize its parameters, demonstrate suboptimal but effective heuristic approximations, and identify numerically stable forms.