Fast and Accurate $k$-means++ via Rejection Sampling

TL;DR

This paper introduces a near-linear time algorithm for $k$-means++ seeding that maintains theoretical guarantees and significantly improves speed while preserving solution quality, making it more practical for large datasets.

Contribution

The paper proposes a novel near-linear time algorithm for $k$-means++ seeding that matches its theoretical guarantees and enhances efficiency over previous methods.

Findings

Algorithm is significantly faster than $k$-means++

Maintains the same theoretical guarantees as $k$-means++

Empirically achieves comparable solution quality

Abstract

$k$-means++ \cite{arthur2007k} is a widely used clustering algorithm that is easy to implement, has nice theoretical guarantees and strong empirical performance. Despite its wide adoption, $k$-means++ sometimes suffers from being slow on large data-sets so a natural question has been to obtain more efficient algorithms with similar guarantees. In this paper, we present a near linear time algorithm for $k$-means++ seeding. Interestingly our algorithm obtains the same theoretical guarantees as $k$-means++ and significantly improves earlier results on fast $k$-means++ seeding. Moreover, we show empirically that our algorithm is significantly faster than $k$-means++ and obtains solutions of equivalent quality.