Random Fourier Features via Fast Surrogate Leverage Weighted Sampling

TL;DR

This paper introduces a faster, simpler method for generating refined random Fourier features using surrogate leverage weighted sampling guided by kernel alignment, improving efficiency in kernel approximation for regression tasks.

Contribution

It proposes a novel surrogate leverage weighted sampling strategy that reduces computational complexity and enhances kernel approximation without matrix inversion, with theoretical guarantees.

Findings

Reduces time complexity from O(ns^2+s^3) to O(ns^2)

Achieves comparable or better prediction performance in kernel ridge regression

Demonstrates efficiency and effectiveness on benchmark datasets

Abstract

In this paper, we propose a fast surrogate leverage weighted sampling strategy to generate refined random Fourier features for kernel approximation. Compared to the current state-of-the-art method that uses the leverage weighted scheme [Li-ICML2019], our new strategy is simpler and more effective. It uses kernel alignment to guide the sampling process and it can avoid the matrix inversion operator when we compute the leverage function. Given n observations and s random features, our strategy can reduce the time complexity from O(ns^2+s^3) to O(ns^2), while achieving comparable (or even slightly better) prediction performance when applied to kernel ridge regression (KRR). In addition, we provide theoretical guarantees on the generalization performance of our approach, and in particular characterize the number of random features required to achieve statistical guarantees in KRR. Experiments on several benchmark datasets demonstrate that our algorithm achieves comparable prediction performance and takes less time cost when compared to [Li-ICML2019].