Many algorithms in scientific computing and data science take advantage of low-rank approximation of matrices and kernels, and understanding why nearly-low-rank structure occurs is essential for their analysis and further development. This paper provides a framework for bounding the best low-rank approximation error of matrices arising from samples of a kernel that is analytically continuable in one of its variables to an open region of the complex plane. Elegantly, the low-rank approximations used in the proof are computable by rational interpolation using the roots and poles of Zolotarev rational functions, leading to a fast algorithm for their construction.

翻译：科学计算与数据科学中的许多算法利用矩阵与核的低秩逼近，理解为何会出现近似低秩结构对其分析与进一步发展至关重要。本文提出一个框架，用于界定由解析可延拓核的样本所生成矩阵的最佳低秩逼近误差——该核在其某一变量上可解析延拓至复平面的开区域。证明过程中使用的低秩逼近可通过佐洛塔廖夫有理函数的零点与极点进行有理插值计算，这一特性优雅地导出了构建此类逼近的快速算法。