Scalable Sparse Subspace Clustering by Orthogonal Matching Pursuit (1507.01238v3)

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.

• The paper introduces SSC-OMP, which employs orthogonal matching pursuit to yield subspace-preserving representations under milder conditions than traditional basis pursuit methods.
• The study demonstrates that SSC-OMP scales efficiently to large datasets, processing up to 100,000 data points with competitive clustering accuracy and reduced computational cost.
• Empirical evaluations on synthetic and real-world datasets validate SSC-OMP's performance, underscoring its potential for real-time applications in computer vision and beyond.

A Technical Review of "Scalable Sparse Subspace Clustering by Orthogonal Matching Pursuit"

The paper "Scalable Sparse Subspace Clustering by Orthogonal Matching Pursuit" by Chong You, Daniel P. Robinson, and René Vidal introduces a novel approach to subspace clustering utilizing orthogonal matching pursuit (OMP) as an alternative to the widely studied basis pursuit (BP) methods. Subspace clustering aims to segment data into clusters that correspond to low-dimensional subspaces, a formulation particularly useful in computer vision tasks such as face clustering and handwritten digit recognition. The key contribution of this work is the demonstration that orthogonal matching pursuit provides a computationally efficient and theoretically robust method for producing subspace-preserving affinities under broad conditions, which substantiates its preferable use in large-scale applications.

Core Methodological Contributions

The authors propose SSC-OMP, building upon the self-expressiveness property of data matrices that lie in a union of subspaces. Traditionally, sparse subspace clustering (SSC) employing ‘1-norm regularization, as championed by SSC-BP, demands solving a large-scale convex optimization problem, which can be computationally intensive. In SSC-OMP, OMP is employed to compute a sparse representation of data points in other data points, ensuring that the solutions adhere to subspace-preservation criteria when certain geometrical and distributional conditions are met.

The paper presents several theoretical advancements:

1. The derivation of conditions under which SSC-OMP guarantees subspace-preserving representations. Notably, these conditions are less stringent than those required for BP methods and are more succinct in their expression.
2. The exploration of both deterministic and random subspace models, revealing that independence and sufficient separation of subspaces, alongside a well-distributed data set, facilitate the realization of subspace-preservation.
3. The realization that SSC-OMP scales effectively with the number of data points, handling up to 100,000 elements efficiently, thereby addressing a key limitation of previous SSC techniques.

Empirical Evaluation

Numerous experiments underscore the performance of SSC-OMP on both synthetic and real-world datasets. In comparison to SSC-BP, SSC-OMP was found to achieve impressive computational gains, being orders of magnitude faster while maintaining competitive clustering performance. This efficiency stems from OMP's greedy algorithmic nature, obviating the need for solving computationally expensive convex programs. On synthetic datasets, SSC-OMP achieves comparable clustering accuracy as the data density increases, albeit with reduced subspace-preserving guarantees compared to SSC-BP.

On real-world datasets, such as MNIST and the Extended Yale B face database, SSC-OMP exhibits robust performance in clustering tasks, generally surpassing methods relying on low-rank or ‘2-norm regularizations. The empirical results suggest that while SSC-BP slightly edges out SSC-OMP with respect to producing perfectly subspace-preserving representations, the scalability and efficiency of SSC-OMP make it highly attractive for large-scale data processing scenarios.

Implications and Future Directions

The theoretical and empirical findings position SSC-OMP as an effective technique for scalable subspace clustering tasks prevalent in various machine learning and computer vision applications. The implications extend beyond mere efficiency, potentially enhancing applications where real-time processing and scaling are critical. The paper leaves room for further exploration in optimizing the trade-off between computational efficiency and clustering accuracy, particularly in the context of real-world data that might deviate from the studied models.

Future inquiries might delve into heterogeneous subspace clustering scenarios, where noise and outlier resilience are essential—areas where SSC-BP has been traditionally strong. Furthermore, extending OMP-based clustering solutions to other domains beyond spectral methods could broaden the applicability and enrich the utility of sparse subspace clustering in more complex, non-Euclidean spaces.

In summary, SSC-OMP introduces a compelling alternative for scalable subspace clustering, presenting important theoretical justifications and demonstrating practical effectiveness in handling large datasets. This paper significantly contributes to the discourse on efficient algorithm design within sparse subspace clustering, inviting further investigation and applications in the broader artificial intelligence domain.