A geometric analysis of subspace clustering with outliers (1112.4258v5)

Abstract: This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower-dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimensions. We develop a novel geometric analysis of an algorithm named sparse subspace clustering (SSC) [In IEEE Conference on Computer Vision and Pattern Recognition, 2009. CVPR 2009 (2009) 2790-2797. IEEE], which significantly broadens the range of problems where it is provably effective. For instance, we show that SSC can recover multiple subspaces, each of dimension comparable to the ambient dimension. We also prove that SSC can correctly cluster data points even when the subspaces of interest intersect. Further, we develop an extension of SSC that succeeds when the data set is corrupted with possibly overwhelmingly many outliers. Underlying our analysis are clear geometric insights, which may bear on other sparse recovery problems. A numerical study complements our theoretical analysis and demonstrates the effectiveness of these methods.

• The paper demonstrates that SSC successfully clusters intersecting subspaces without requiring strict angle conditions.
• It proves that SSC remains robust against overwhelming outliers by accurately isolating them from true data points.
• The study employs geometric functional analysis to extend SSC’s effectiveness to high-dimensional settings with intersecting subspaces.

A Geometric Analysis of Subspace Clustering with Outliers

In the paper "A geometric analysis of subspace clustering with outliers," the authors, Mahdi Soltanolkotabi and Emmanuel J. Candès, undertake a comprehensive paper of Sparse Subspace Clustering (SSC), with an emphasis on its efficacy in the presence of outliers. Subspace clustering is a critical unsupervised learning problem wherein data points lie close to multiple low-dimensional subspaces rather than a single low-dimensional plane. The challenge is compounded when these subspaces intersect or when the data is contaminated with outliers. This paper provides a novel geometric perspective to SSC, enhancing its theoretical foundation and practical applicability.

Contributions and Theoretical Insights

The primary contribution of this work lies in its geometric insights into the problem of clustering data drawn from a union of multiple subspaces. The authors decisively extend the theoretical boundaries of SSC, proving that it can successfully cluster subspaces of dimensionality comparable to the ambient space dimension. This is achievable even when the subspaces intersect, a notable advancement beyond previous assumptions.

Four key theoretical insights are presented:

1. Subspace Detection with Intersecting Subspaces: The authors demonstrate that SSC can correctly cluster data points even when subspaces intersect, without requiring minimum angle conditions between subspaces. This represents a significant relaxation of previously stringent conditions necessary for clustering success.
2. Handling High-Dimensional Subspaces: The paper proves that SSC is effective for subspaces with dimensions close to the ambient dimension, under the condition that the number of points per subspace scales suitably with the dimension.
3. Robustness Against Outliers: A significant extension of SSC is presented, which is provably robust in the face of overwhelming numbers of outliers. The proposed method accurately isolates outliers even when their number far exceeds that of the actual data points.
4. Geometric Framework: Employing geometric functional analysis, the authors provide a clear geometric framework for understanding when SSC will succeed. This framework could be beneficial for addressing other sparse recovery challenges.

Empirical Evaluation

The theoretical insights are fortified by numerical experiments demonstrating SSC’s robustness and accuracy under various scenarios, including high-dimensional settings and data contaminated with substantial noise or outliers. These experiments validate the analytical results, showing a small gap between theoretical predictions and practical performance.

Practical Implications

The implications of these findings are considerable, particularly in fields reliant on unsupervised learning and computer vision. The capability to cluster intersecting subspaces broadens the applicability of SSC in practical scenarios where data structure often defies simpler assumptions. Moreover, the robust handling of outliers facilitates cleaner and more accurate data clustering, crucial for applications like motion segmentation in videos or disease detection in large-scale medical datasets.

Future Directions

The paper's insights suggest several avenues for future research:

• Noisy Data Frameworks: Expanding the analysis to noisy subspace clustering to establish more comprehensive solutions under real-world conditions.
• Sparse Recovery Problems: Applying the geometric insights offered here to a broader class of sparse recovery problems beyond subspace clustering.
• Algorithmic Enhancements: Developing more efficient computational techniques inspired by the theoretical advancements for real-time applications.

In conclusion, this paper provides a substantial enhancement to the theoretical and practical understanding of subspace clustering, particularly in challenging real-world scenarios involving data intersections and pervasive outliers. The rigorous geometric analysis employed herein not only fortifies the SSC methodology but also opens avenues for novel applications and problem-solving strategies within the field of data science and machine learning.