Revisiting the Nystrom Method for Improved Large-Scale Machine Learning

TL;DR

This paper empirically evaluates randomized low-rank approximation algorithms for SPSD matrices, compares sampling and projection methods, and provides improved theoretical bounds to guide large-scale machine learning applications.

Contribution

It offers a comprehensive empirical comparison of sampling and projection methods and introduces improved theoretical bounds for these algorithms.

Findings

Sampling and projection methods have complementary strengths.

Data preprocessing significantly affects algorithm performance.

Existing theory poorly predicts practical performance.

Abstract

We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results consist of an empirical evaluation of the performance quality and running time of sampling and projection methods on a diverse suite of SPSD matrices. Our results highlight complementary aspects of sampling versus projection methods; they characterize the effects of common data preprocessing steps on the performance of these algorithms; and they point to important differences between uniform sampling and nonuniform sampling methods based on leverage scores. In addition, our empirical results illustrate that existing theory is so weak that it does not provide even a qualitative guide to practice. Thus, we complement our empirical results with a suite of worst-case theoretical bounds for both random sampling and random projection methods. These bounds are qualitatively superior to existing bounds---e.g. improved additive-error bounds for spectral and Frobenius norm error and relative-error bounds for trace norm error---and they point to future directions to make these algorithms useful in even larger-scale machine learning applications.