Sparse Subspace Clustering: Algorithm, Theory, and Applications (1203.1005v3)

Abstract: In many real-world problems, we are dealing with collections of high-dimensional data, such as images, videos, text and web documents, DNA microarray data, and more. Often, high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories the data belongs to. In this paper, we propose and study an algorithm, called Sparse Subspace Clustering (SSC), to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of data into subspaces. Since solving the sparse optimization program is in general NP-hard, we consider a convex relaxation and show that, under appropriate conditions on the arrangement of subspaces and the distribution of data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm can be solved efficiently and can handle data points near the intersections of subspaces. Another key advantage of the proposed algorithm with respect to the state of the art is that it can deal with data nuisances, such as noise, sparse outlying entries, and missing entries, directly by incorporating the model of the data into the sparse optimization program. We demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real-world problems of motion segmentation and face clustering.

• The paper introduces the SSC algorithm that leverages sparse representations to accurately recover low-dimensional subspaces from high-dimensional data.
• It integrates ℓ₁-minimization with spectral clustering to robustly handle noise, outliers, and missing data, ensuring effective subspace recovery.
• Empirical results on motion segmentation and face clustering tasks demonstrate SSC’s lower error rates and practical advantages over competing methods.

Sparse Subspace Clustering: Algorithm, Theory, and Applications

The paper "Sparse Subspace Clustering: Algorithm, Theory, and Applications" by Ehsan Elhamifar and Rene Vidal introduces the Sparse Subspace Clustering (SSC) algorithm, a method designed for efficiently clustering high-dimensional data points that lie in a union of low-dimensional subspaces. The paper establishes SSC both in terms of theoretical underpinnings and practical applications, demonstrating its versatility and robustness in handling noisy, incomplete, and corrupted data.

Key Contributions

1. Sparse Representation for Clustering: The central tenet of SSC hinges on the concept of obtaining a sparse representation of a given data point in terms of other data points. The algorithm posits that a sparse representation naturally aligns data points from the same subspace. This sparse representation is then utilized within a spectral clustering framework to partition the data into distinct subspaces.
2. Handling Data Nuisances: Unlike existing methods, SSC is adept at dealing with various data imperfections such as noise, sparse outliers, and missing entries. These capabilities are incorporated directly into the sparse optimization program, which leverages ℓ₁-minimization to recover the desired sparse representations.
3. Theoretical Guarantees: The paper extends sparse representation theory to the multi-subspace setting, providing conditions under which SSC guarantees correct subspace recovery. It does so notably under both independent and disjoint subspace models. For independent subspaces, SSC consistently recovers subspace-sparse representations as long as the dimensions of individual subspaces sum to the dimension of the ambient space. For disjoint subspaces, the theory outlines conditions involving principal angles and data distribution ensuring the success of SSC.
4. Algorithmic Innovations: SSC is implemented using an efficient Alternating Direction Method of Multipliers (ADMM) approach, making it viable for large-scale problems. This implementation significantly improves over general interior point solvers, reducing computational overhead while maintaining robustness.

Practical Applications

• Motion Segmentation: One of the real-world applications of SSC detailed in the paper is motion segmentation in video sequences. SSC successfully segments trajectories of feature points from multiple rigidly moving objects into their respective motions. The algorithm outperforms other state-of-the-art techniques in the Hopkins 155 dataset.
• Face Clustering: Another application area is clustering face images under varying illumination conditions. The Extended Yale B dataset experiments highlight SSC's effectiveness in handling face images that lie in a union of low-dimensional subspaces, even when the images are corrupted by noise and occlusions.

Numerical Results

The numerical experiments conducted on synthetic and real datasets corroborate the theoretical claims. In the Hopkins 155 motion segmentation dataset, SSC achieves significantly lower clustering errors compared to methods like Local Subspace Affinity (LSA), Spectral Curvature Clustering (SCC), and Low-Rank Representation (LRR). Notably, SSC remains robust across different pre-processing conditions, such as using raw 2F-dimensional trajectories or their PCA projections.

Similarly, in the face clustering task using the Extended Yale B dataset, SSC demonstrates superior performance, especially when handling data with sparse outlying entries, underscoring the practical utility of incorporating data corruption models directly into the sparse optimization framework.

Theoretical Implications

The theoretical contributions of SSC extend sparse representation theory to encompass the multi-subspace setting under both independent and disjoint subspace models. The theoretical results are pioneering in demonstrating conditions for guaranteed subspace-sparse recovery, particularly leveraging principal angles and subspace intersections to provide practical recovery bounds. These insights deepen the understanding of clustering in high-dimensional spaces and pave the way for further research into subspace clustering methodologies.

Future Directions

Future work could explore several avenues:

• Extension to Nonlinear Subspaces: Investigating the potential of SSC in scenarios where data points lie in a union of nonlinear subspaces.
• Scalability Enhancements: Further optimizing the ADMM implementation or exploring alternative optimization approaches to enhance scalability.
• Robustness Analysis: Providing a more detailed theoretical analysis of SSC's robustness in noisy and missing data conditions, potentially incorporating probabilistic models.

Conclusion

SSC stands out as a significant advancement in the clustering of high-dimensional data, particularly through its novel use of sparse representations and robust optimization techniques. Its strong theoretical foundations, combined with practical efficacy in domains such as motion segmentation and face clustering, make it a valuable tool for researchers and practitioners dealing with complex, real-world datasets.