Oracle Based Active Set Algorithm for Scalable Elastic Net Subspace Clustering (1605.02633v1)

Abstract: State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with $\ell_1$, $\ell_2$ or nuclear norms. $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad theoretical conditions, but the clusters may not be connected. $\ell_2$ and nuclear norm regularization often improve connectivity, but give a subspace-preserving affinity only for independent subspaces. Mixed $\ell_1$, $\ell_2$ and nuclear norm regularizations offer a balance between the subspace-preserving and connectedness properties, but this comes at the cost of increased computational complexity. This paper studies the geometry of the elastic net regularizer (a mixture of the $\ell_1$ and $\ell_2$ norms) and uses it to derive a provably correct and scalable active set method for finding the optimal coefficients. Our geometric analysis also provides a theoretical justification and a geometric interpretation for the balance between the connectedness (due to $\ell_2$ regularization) and subspace-preserving (due to $\ell_1$ regularization) properties for elastic net subspace clustering. Our experiments show that the proposed active set method not only achieves state-of-the-art clustering performance, but also efficiently handles large-scale datasets.

• The paper introduces the ORGEN algorithm that leverages an oracle-guided active set method to balance sparsity and connectivity for efficient subspace clustering.
• Theoretical analysis provides conditions for achieving subspace-preserving affinity matrices by tuning the λ parameter in elastic net regularization.
• Experimental results on datasets like MNIST and CovType demonstrate over 93% accuracy and significant computational efficiency improvements over existing methods.

Elastic Net Subspace Clustering: Theory and Algorithmic Advances

The paper "Elastic Net Subspace Clustering: Theoretical Analysis and an Oracle Based Active Set Algorithm" presents a comprehensive investigation into the application of elastic net regularization for subspace clustering. A fundamental problem in computer vision and data analysis, subspace clustering aims to identify subspaces within high-dimensional data. This task is critical for applications such as motion segmentation, face clustering, and image representation.

Main Contributions

The paper's primary contribution lies in leveraging the elastic net, a combination of $\ell_1$ and $\ell_2$ regularization, to address the balance between accuracy (subspace preservation) and connectivity (inter-point connectedness) within clustering affinity graphs. The elastic net approach enables a nuanced trade-off between sparsity and dense representation of data points, crucial for improving clustering outcomes.

The authors introduce a provably correct and scalable active set algorithm, referred to as ORacle Guided Elastic Net solver (ORGEN). This algorithm effectively solves the elastic net problem by capitalizing on the geometric structure of the elastic net solution. The novelty of the approach stems from identifying an oracle region in the parameter space and utilizing it to inform efficient updates to the active set during optimization, ensuring convergence to the optimal solution.

Theoretical Insights

The paper provides rigorous theoretical insights into the conditions under which the elastic net yields subspace-preserving solutions. A key theoretical contribution is the geometric interpretation of the oracle point and region, providing a clear framework for understanding the trade-off between connectedness and subspace preservation based on the parameter $\lambda,$ controlling the mix of $\ell_1$ and $\ell_2$ norms.

The authors introduce conditions that guarantee subspace preservation of the affinity matrix, improving on prior work by deriving results under local data distributions—capturing the behavior of the data distribution in a more granular manner than previous global approaches. These results culminate in the condition that sparsity and connectedness can be systematically balanced by tuning $\lambda$.

Experimental Evaluation

The effectiveness of the proposed active set algorithm is demonstrated through comprehensive experiments on synthetic and real-world datasets, including large-scale datasets not previously manageable with other subspace clustering methods. The proposed method not only outperformed existing methods in terms of clustering accuracy but also demonstrated significant improvements in computational efficiency, which is critical for handling large datasets.

For example, evaluations on the MNIST dataset with 70,000 samples showed a clustering accuracy surpassing 93% with a running time significantly reduced compared to standard methods. On the CovType dataset, encompassing over half a million samples, the algorithm efficiently provided accurate clustering where other methods either failed or were computationally prohibitive.

Implications and Future Work

This work advances the current understanding of subspace clustering using elastic net regularization by providing an efficient algorithmic solution paired with robust theoretical underpinnings. It invites further exploration into the scalability of subspace clustering methods, especially as datasets continue to grow in size and complexity.

Future developments could consider extensions of this work to other forms of regularization or apply similar active set methods for other convex and non-convex optimization problems in machine learning. Moreover, the interplay between oracle-based approaches and deep learning could open new pathways for subspace learning in neural networks, potentially leading to more interpretable and flexible architectures.

In conclusion, the authors have significantly advanced the field of subspace clustering with a theoretically sound and computationally efficient method, setting a strong foundation for future research and practical applications.