Making the Nyström method highly accurate for low-rank approximations
This work addresses the need for high-accuracy kernel matrix approximations in machine learning and data analysis, though it is incremental as it builds on existing Nyström methods with heuristic refinements.
The authors tackled the problem of improving the accuracy of the Nyström method for low-rank approximations of nonsymmetric and/or rectangular matrices, achieving results that sometimes approach SVD quality with small numbers of sampling steps.
The Nyström 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 Nyström method reach high accuracies for nonsymmetric and/or rectangular matrices. The resulting methods (called high-accuracy Nyström methods) treat the Nyström 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.