# Subspace Clustering using Ensembles of K-Subspaces

arXiv: https://arxiv.org/abs/1709.04744 · HTML: https://mlsanity.com/paper/1709.04744/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2017.

## Built on by (most cited first, 7 of 7)

- [Unsupervised Manifold Linearizing and Clustering](https://mlsanity.com/paper/2301.01805/) (2023)
- [Convergence and Recovery Guarantees of the K-Subspaces Method for Subspace Clustering](https://mlsanity.com/paper/2206.05553/) (2022)
- [Adversarial Subspace Generation for Outlier Detection in High-Dimensional Data](https://mlsanity.com/paper/2504.07522/) (2025)
- [PMSSC: Parallelizable multi-subset based self-expressive model for subspace clustering](https://mlsanity.com/paper/2111.12232/) (2021)
- [ALPCAHUS: Subspace Clustering for Heteroscedastic Data](https://mlsanity.com/paper/2505.18918/) (2025)
- [Provable Clustering of a Union of Linear Manifolds Using Optimal Directions](https://mlsanity.com/paper/2201.02745/) (2022)
- [Robust Multi-Manifold Clustering via Simplex Paths](https://mlsanity.com/paper/2507.10710/) (2025)

## Abstract (arXiv)

Subspace clustering is the unsupervised grouping of points lying near a union of low-dimensional linear subspaces. Algorithms based directly on geometric properties of such data tend to either provide poor empirical performance, lack theoretical guarantees, or depend heavily on their initialization. We present a novel geometric approach to the subspace clustering problem that leverages ensembles of the K-subspaces (KSS) algorithm via the evidence accumulation clustering framework. Our algorithm, referred to as ensemble K-subspaces (EKSS), forms a co-association matrix whose (i,j)th entry is the number of times points i and j are clustered together by several runs of KSS with random initializations. We prove general recovery guarantees for any algorithm that forms an affinity matrix with entries close to a monotonic transformation of pairwise absolute inner products. We then show that a specific instance of EKSS results in an affinity matrix with entries of this form, and hence our proposed algorithm can provably recover subspaces under similar conditions to state-of-the-art algorithms. The finding is, to the best of our knowledge, the first recovery guarantee for evidence accumulation clustering and for KSS variants. We show on synthetic data that our method performs well in the traditionally challenging settings of subspaces with large intersection, subspaces with small principal angles, and noisy data. Finally, we evaluate our algorithm on six common benchmark datasets and show that unlike existing methods, EKSS achieves excellent empirical performance when there are both a small and large number of points per subspace.