Fast Statistical Leverage Score Approximation in Kernel Ridge Regression

TL;DR

This paper introduces a fast, linear-time algorithm for accurately approximating statistical leverage scores in kernel ridge regression, significantly improving the efficiency of Nyström approximation without sacrificing prediction accuracy.

Contribution

It provides a novel analytic formula for leverage scores in stationary-kernel KRR, enabling efficient sampling for Nyström approximation with theoretical guarantees.

Findings

Method is orders of magnitude faster than existing approaches.

Maintains the same prediction accuracy as traditional methods.

Theoretical analysis confirms the accuracy of leverage score approximation.

Abstract

Nystr\"om approximation is a fast randomized method that rapidly solves kernel ridge regression (KRR) problems through sub-sampling the n-by-n empirical kernel matrix appearing in the objective function. However, the performance of such a sub-sampling method heavily relies on correctly estimating the statistical leverage scores for forming the sampling distribution, which can be as costly as solving the original KRR. In this work, we propose a linear time (modulo poly-log terms) algorithm to accurately approximate the statistical leverage scores in the stationary-kernel-based KRR with theoretical guarantees. Particularly, by analyzing the first-order condition of the KRR objective, we derive an analytic formula, which depends on both the input distribution and the spectral density of stationary kernels, for capturing the non-uniformity of the statistical leverage scores. Numerical experiments demonstrate that with the same prediction accuracy our method is orders of magnitude more efficient than existing methods in selecting the representative sub-samples in the Nystr\"om approximation.