The Nystr\"om method is a convenient heuristic method to obtain low-rank approximations to kernel matrices in nearly linear complexity. Existing studies typically use the method to approximate positive semidefinite matrices with low or modest accuracies. In this work, we propose a series of heuristic strategies to make the Nystr\"om method reach high accuracies for nonsymmetric and/or rectangular matrices. The resulting methods (called high-accuracy Nystr\"om methods) treat the Nystr\"om method and a skinny rank-revealing factorization as a fast pivoting strategy in a progressive alternating direction refinement process. Two refinement mechanisms are used: alternating the row and column pivoting starting from a small set of randomly chosen columns, and adaptively increasing the number of samples until a desired rank or accuracy is reached. A fast subset update strategy based on the progressive sampling of Schur complements is further proposed to accelerate the refinement process. Efficient randomized accuracy control is also provided. Relevant accuracy and singular value analysis is given to support some of the heuristics. Extensive tests with various kernel functions and data sets show how the methods can quickly reach prespecified high accuracies in practice, sometimes with quality close to SVDs, using only small numbers of progressive sampling steps.

翻译：Nyström方法是一种方便的启发式方法，可以在近线性复杂度下获得核矩阵的低秩逼近。现有研究通常使用该方法对半正定矩阵进行低精度或中等精度的逼近。本文提出一系列启发式策略，使Nyström方法能够对非对称和/或矩形矩阵达到高精度。所提出的方法（称为高精度Nyström方法）将Nyström方法与瘦秩揭示分解作为渐进交替方向细化过程中的快速枢轴选择策略。采用两种细化机制：从随机选择的少量列开始交替进行行和列枢轴选择，并自适应增加样本数量直至达到所需秩或精度。进一步提出基于Schur补渐进采样的快速子集更新策略以加速细化过程，同时提供高效的随机化精度控制。相关精度与奇异值分析为部分启发式策略提供了理论支持。针对多种核函数与数据集的广泛测试表明，该方法仅需少量渐进采样步骤即可快速达到预定的高精度，有时可接近SVD的质量。