Sampling-based Nystr\"om Approximation and Kernel Quadrature

TL;DR

This paper provides improved error bounds for Nyström kernel approximation, introduces a refined subspace selection method with theoretical guarantees, and explores applications to kernel quadrature with new insights and numerical validation.

Contribution

It presents an improved error bound for Nyström approximation, a new subspace selection method with theoretical guarantees, and applies these to kernel quadrature with novel theoretical and numerical results.

Findings

Enhanced error bounds for Nyström approximation with i.i.d. sampling.

A refined subspace selection method with theoretical guarantees for non-i.i.d. points.

New theoretical guarantees and numerical results for kernel quadrature.

Abstract

We analyze the Nystr\"om approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nystr\"om approximation with i.i.d. sampling and singular-value decomposition in the continuous regime; the proof techniques are borrowed from statistical learning theory. We further introduce a refined selection of subspaces in Nystr\"om approximation with theoretical guarantees that is applicable to non-i.i.d. landmark points. Finally, we discuss their application to convex kernel quadrature and give novel theoretical guarantees as well as numerical observations.