# k-Means Clustering Is Matrix Factorization

arXiv: https://arxiv.org/abs/1512.07548 · HTML: https://mlsanity.com/paper/1512.07548/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Dec 2015.

## Built on by (most cited first, 11 of 11)

- [GCFAgg: Global and Cross-view Feature Aggregation for Multi-view Clustering](https://mlsanity.com/paper/2305.06799/) (2023)
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- [Cut Less, Fold More: Model Compression through the Lens of Projection Geometry](https://mlsanity.com/paper/2602.18116/) (2026)
- [Trusted Mamba Contrastive Network for Multi-View Clustering](https://mlsanity.com/paper/2412.16487/) (2024)
- [A Restarted Large-Scale Spectral Clustering with Self-Guiding and Block Diagonal Representation](https://mlsanity.com/paper/2306.15138/) (2023)
- [Improving Spectral Clustering Using Spectrum-Preserving Node Aggregation](https://mlsanity.com/paper/2110.12328/) (2021)
- [Deep Contrastive Multi-view Clustering under Semantic Feature Guidance](https://mlsanity.com/paper/2403.05768/) (2024)
- [Vocabulary-free few-shot learning for Vision-Language Models](https://mlsanity.com/paper/2506.04005/) (2025)
- [Binary Orthogonal Non-negative Matrix Factorization](https://mlsanity.com/paper/2210.10660/) (2022)
- [On the Global Solution of Soft k-Means](https://mlsanity.com/paper/2212.03589/) (2022)
- [A Unified Matrix Factorization Framework for Classical and Robust Clustering](https://mlsanity.com/paper/2510.21172/) (2025)

## Abstract (arXiv)

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.