Randomized Clustered Nystrom for Large-Scale Kernel Machines

TL;DR

This paper introduces a scalable randomized Nystrom method that improves low-rank kernel matrix approximation for large-scale, high-dimensional data by combining optimal landmark selection with efficient clustering.

Contribution

It presents a novel algorithm for optimal Nystrom approximation with landmark points exceeding target rank and a scalable randomized landmark selection method using low-dimensional projections.

Findings

The method achieves competitive approximation accuracy.

It significantly reduces computational costs for high-dimensional data.

Experiments validate the efficiency and effectiveness of the approach.

Abstract

The Nystrom method has been popular for generating the low-rank approximation of kernel matrices that arise in many machine learning problems. The approximation quality of the Nystrom method depends crucially on the number of selected landmark points and the selection procedure. In this paper, we present a novel algorithm to compute the optimal Nystrom low-approximation when the number of landmark points exceed the target rank. Moreover, we introduce a randomized algorithm for generating landmark points that is scalable to large-scale data sets. The proposed method performs K-means clustering on low-dimensional random projections of a data set and, thus, leads to significant savings for high-dimensional data sets. Our theoretical results characterize the tradeoffs between the accuracy and efficiency of our proposed method. Extensive experiments demonstrate the competitive performance as well as the efficiency of our proposed method.