Nystr\"om Kernel Mean Embeddings

TL;DR

This paper introduces a Nyström-based approximation method for kernel mean embeddings that reduces computational costs while maintaining statistical accuracy, enabling scalable distribution comparison and integration.

Contribution

It presents a novel Nyström approximation technique for kernel mean embeddings with theoretical error bounds and practical applications.

Findings

Achieves $n^{-1/2}$ convergence rate with smaller samples

Reduces computational complexity for large datasets

Demonstrates effectiveness through numerical experiments

Abstract

Kernel mean embeddings are a powerful tool to represent probability distributions over arbitrary spaces as single points in a Hilbert space. Yet, the cost of computing and storing such embeddings prohibits their direct use in large-scale settings. We propose an efficient approximation procedure based on the Nystr\"om method, which exploits a small random subset of the dataset. Our main result is an upper bound on the approximation error of this procedure. It yields sufficient conditions on the subsample size to obtain the standard $n^{-1/2}$ rate while reducing computational costs. We discuss applications of this result for the approximation of the maximum mean discrepancy and quadrature rules, and illustrate our theoretical findings with numerical experiments.