The spectral norm error of the naive Nystrom extension

TL;DR

This paper establishes the first relative-error spectral norm bound for the naive Nystrom extension, connecting it to column subset selection and utilizing a matrix Chernoff bound for sampling without replacement.

Contribution

It provides a novel theoretical bound on the spectral norm error of the naive Nystrom extension, linking it to the column subset selection problem.

Findings

First relative-error spectral norm bound for naive Nystrom extension

Connection established between Nystrom extension and column subset selection

Utilizes matrix Chernoff bound for sampling without replacement

Abstract

The naive Nystrom extension forms a low-rank approximation to a positive-semidefinite matrix by uniformly randomly sampling from its columns. This paper provides the first relative-error bound on the spectral norm error incurred in this process. This bound follows from a natural connection between the Nystrom extension and the column subset selection problem. The main tool is a matrix Chernoff bound for sampling without replacement.