On the Consistency of $k$-means++ algorithm

TL;DR

This paper proves that the expected clustering cost of the k-means++ algorithm on samples converges to the population expected value, enabling reliable subsampling for large datasets.

Contribution

It establishes the convergence of the sample-based k-means++ objective to the population objective, supporting its use in large-scale clustering.

Findings

Expected value of k-means++ objective converges to the population value

Sample-based approximation maintains constant factor guarantees

Supports subsampling for large datasets

Abstract

We prove in this paper that the expected value of the objective function of the $k$-means++ algorithm for samples converges to population expected value. As $k$-means++, for samples, provides with constant factor approximation for $k$-means objectives, such an approximation can be achieved for the population with increase of the sample size. This result is of potential practical relevance when one is considering using subsampling when clustering large data sets (large data bases).