Greedy Subspace Clustering

TL;DR

This paper introduces simple, efficient algorithms for subspace clustering that guarantee exact performance under weaker conditions, demonstrating strong results on synthetic and real-world data with lower computational costs.

Contribution

The paper proposes new algorithms for subspace clustering with theoretical guarantees and practical efficiency, outperforming existing methods in simplicity and computational cost.

Findings

Guaranteed exact clustering under weaker conditions

Competitive performance on motion segmentation and face clustering

Lower computational cost compared to state-of-the-art methods

Abstract

We consider the problem of subspace clustering: given points that lie on or near the union of many low-dimensional linear subspaces, recover the subspaces. To this end, one first identifies sets of points close to the same subspace and uses the sets to estimate the subspaces. As the geometric structure of the clusters (linear subspaces) forbids proper performance of general distance based approaches such as K-means, many model-specific methods have been proposed. In this paper, we provide new simple and efficient algorithms for this problem. Our statistical analysis shows that the algorithms are guaranteed exact (perfect) clustering performance under certain conditions on the number of points and the affinity between subspaces. These conditions are weaker than those considered in the standard statistical literature. Experimental results on synthetic data generated from the standard unions of subspaces model demonstrate our theory. We also show that our algorithm performs competitively against state-of-the-art algorithms on real-world applications such as motion segmentation and face clustering, with much simpler implementation and lower computational cost.