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：在科学计算与数据科学的诸多算法中，矩阵与核的低秩逼近技术被广泛应用，理解近低秩结构产生的原因对于算法分析与进一步发展至关重要。本文提出了一个理论框架，用于界定由解析核采样所生成矩阵的最佳低秩逼近误差，其中要求核函数在其某一变量上可解析延拓至复平面的某个开区域。证明过程中所采用的低秩逼近可通过有理插值方法计算得出，该插值以佐洛塔廖夫有理函数的零点与极点为节点，进而导出了一个可快速构建此类逼近的高效算法。