k-Means Clustering Is Matrix Factorization

TL;DR

This paper demonstrates that k-means clustering can be formulated as a matrix factorization problem by expressing its objective as a Frobenius norm of a data matrix and its low-rank approximation.

Contribution

It explicitly shows the equivalence between k-means clustering and matrix factorization, clarifying a connection often implied but not formally stated in literature.

Findings

k-means objective equals Frobenius norm of data matrix minus low-rank approximation

Provides a formal link between clustering and matrix factorization

Clarifies a common but implicit understanding in the literature

Abstract

We show that the objective function of conventional k-means clustering can be expressed as the Frobenius norm of the difference of a data matrix and a low rank approximation of that data matrix. In short, we show that k-means clustering is a matrix factorization problem. These notes are meant as a reference and intended to provide a guided tour towards a result that is often mentioned but seldom made explicit in the literature.