Greedy Feature Selection for Subspace Clustering (1303.4778v2)

Abstract: Unions of subspaces provide a powerful generalization to linear subspace models for collections of high-dimensional data. To learn a union of subspaces from a collection of data, sets of signals in the collection that belong to the same subspace must be identified in order to obtain accurate estimates of the subspace structures present in the data. Recently, sparse recovery methods have been shown to provide a provable and robust strategy for exact feature selection (EFS)--recovering subsets of points from the ensemble that live in the same subspace. In parallel with recent studies of EFS with L1-minimization, in this paper, we develop sufficient conditions for EFS with a greedy method for sparse signal recovery known as orthogonal matching pursuit (OMP). Following our analysis, we provide an empirical study of feature selection strategies for signals living on unions of subspaces and characterize the gap between sparse recovery methods and nearest neighbor (NN)-based approaches. In particular, we demonstrate that sparse recovery methods provide significant advantages over NN methods and the gap between the two approaches is particularly pronounced when the sampling of subspaces in the dataset is sparse. Our results suggest that OMP may be employed to reliably recover exact feature sets in a number of regimes where NN approaches fail to reveal the subspace membership of points in the ensemble.

• The paper demonstrates that greedy feature selection via Orthogonal Matching Pursuit achieves exact feature selection in subspace clustering.
• It establishes necessary conditions on mutual coherence and covering radius to ensure precise feature recovery compared to NN-based methods.
• Empirical analysis indicates that greedy methods can significantly outperform traditional nearest neighbor approaches in high-dimensional, sparse sampling scenarios.

An Exploration into Greedy Feature Selection for Subspace Clustering

The research paper "Greedy Feature Selection for Subspace Clustering" by Dyer, Sankaranarayanan, and Baraniuk presents a detailed analysis of subspace clustering by leveraging greedy feature selection strategies. This paper is positioned within the broader context of handling high-dimensional and heterogeneous data by exploiting its intrinsic low-dimensional geometric structure.

Unions of Subspaces and Their Importance

Subspace clustering is pivotal in data analysis where observations lie in a union of subspaces of unknown dimensions. This approach extends linear subspace models which are commonly utilized in applications such as machine learning and signal processing. Principal among these techniques is Principal Component Analysis (PCA) that provides computational efficiency for low-rank data approximation. However, the paper emphasizes situations where a single subspace model is insufficient, and a union of subspaces captures more complexity, necessary for applications such as image and signal processing.

Feature Selection in Subspace Clustering

A critical challenge in subspace clustering involves identifying subspaces and selecting features that belong to the same subspace. Traditional methods, which rely on nearest neighbor (NN)-based feature selection, often fail when the data points are sampled sparsely or when subspaces intersect significantly. This inadequacy underscores the need for methods that ensure exact feature selection (EFS).

Advances through Sparse Recovery Methods

The paper contributes significantly by exploring greedy methods like Orthogonal Matching Pursuit (OMP) for achieving EFS in sparse recovery, a shift from more established approaches like $\ell_1$-minimization. A comparison is drawn between greedy strategies and sparse recovery methods in handling challenges characteristic of subspace clustering, where traditional NN methods fall short.

Theoretical Contributions and Empirical Findings

The authors develop necessary conditions under which OMP gives exact feature selection. The critical determinant of success in these methods is the relationship between subspace interaction—through conceptions such as mutual coherence and covering radius—and precise signal recovery. The mutual coherence provides a measure of similarity or dependency between points of different subspaces, while the covering radius encapsulates how well these subspaces are sampled.

Significantly, their empirical paper reveals the conditions and limitations under which greedy methods outperform nearest neighbors. Sparse recovery methods, especially under sparse subspace sampling, are sometimes shown to dramatically outperform NN approaches.

Implications for Future Research

This paper contributes to the understanding of sparse recovery in subspace clustering, suggesting that reduced dimensional subspace models can have superior performance in specific regimes. The implications extend to potential applications in compressed sensing and dictionary learning, particularly in determining the structural composition of adaptive systems over time. Furthermore, the insights into mutual coherence and covering radius foster opportunities to develop new algorithms that exploit intrinsic data structure for better feature selection and clustering.

Conclusion

The authors offer a significant advancement in subspace clustering through the lens of sparse recovery, positioning greedy methods as viable and competitive alternatives to traditional techniques. This work invites further exploration into feature selection mechanisms, focusing on adaptive models to capitalize on structured sparsity. With ongoing challenges in high-dimensional data analysis, the paper situates its findings for impactful application across diverse domains, prompting a reevaluation of existing models and encouraging new, informed strategies in feature selection and inference.