Making the Nystr\"om method highly accurate for low-rank approximations

TL;DR

This paper introduces high-accuracy Nystr"om methods that significantly improve low-rank kernel matrix approximations, achieving near-SVD quality efficiently through iterative refinement and adaptive sampling strategies.

Contribution

The paper develops heuristic strategies that enhance the Nystr"om method's accuracy for nonsymmetric and rectangular matrices using progressive refinement and adaptive sampling.

Findings

Methods reach high accuracy close to SVDs

Efficiently handle various kernel functions and data sets

Achieve high-quality approximations with few sampling steps

Abstract

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.