Preconditioned Data Sparsification for Big Data with Applications to PCA and K-means

TL;DR

This paper introduces a randomized preconditioning and sampling scheme for compressing large datasets, enabling faster PCA and K-means clustering with theoretical guarantees and practical benefits.

Contribution

It proposes a novel data sparsification method that combines randomized preconditioning with sampling, improving efficiency and providing theoretical error bounds for PCA and K-means.

Findings

The method achieves near-tight theoretical guarantees for PCA and K-means.

Numerical experiments show significant speedups and accuracy benefits on standard datasets.

The approach outperforms some existing sampling techniques in practical scenarios.

Abstract

We analyze a compression scheme for large data sets that randomly keeps a small percentage of the components of each data sample. The benefit is that the output is a sparse matrix and therefore subsequent processing, such as PCA or K-means, is significantly faster, especially in a distributed-data setting. Furthermore, the sampling is single-pass and applicable to streaming data. The sampling mechanism is a variant of previous methods proposed in the literature combined with a randomized preconditioning to smooth the data. We provide guarantees for PCA in terms of the covariance matrix, and guarantees for K-means in terms of the error in the center estimators at a given step. We present numerical evidence to show both that our bounds are nearly tight and that our algorithms provide a real benefit when applied to standard test data sets, as well as providing certain benefits over related sampling approaches.