Scalable Sparse Subspace Clustering by Orthogonal Matching Pursuit

TL;DR

This paper introduces a scalable subspace clustering method using orthogonal matching pursuit, which is both computationally efficient and guarantees subspace-preserving affinity under broad conditions, outperforming traditional regularization-based methods.

Contribution

It proposes a novel subspace clustering approach based on orthogonal matching pursuit that balances efficiency and theoretical guarantees, unlike existing regularization methods.

Findings

Method is computationally efficient.

Guarantees subspace-preserving affinity under broad conditions.

Achieves best trade-off between accuracy and efficiency in experiments.

Abstract

Subspace clustering methods based on $\ell_1$, $\ell_2$ or nuclear norm regularization have become very popular due to their simplicity, theoretical guarantees and empirical success. However, the choice of the regularizer can greatly impact both theory and practice. For instance, $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad conditions (e.g., arbitrary subspaces and corrupted data). However, it requires solving a large scale convex optimization problem. On the other hand, $\ell_2$ and nuclear norm regularization provide efficient closed form solutions, but require very strong assumptions to guarantee a subspace-preserving affinity, e.g., independent subspaces and uncorrupted data. In this paper we study a subspace clustering method based on orthogonal matching pursuit. We show that the method is both computationally efficient and guaranteed to give a subspace-preserving affinity under broad conditions. Experiments on synthetic data verify our theoretical analysis, and applications in handwritten digit and face clustering show that our approach achieves the best trade off between accuracy and efficiency.