Robust and Efficient Subspace Segmentation via Least Squares Regression (1404.6736v1)

Abstract: This paper studies the subspace segmentation problem which aims to segment data drawn from a union of multiple linear subspaces. Recent works by using sparse representation, low rank representation and their extensions attract much attention. If the subspaces from which the data drawn are independent or orthogonal, they are able to obtain a block diagonal affinity matrix, which usually leads to a correct segmentation. The main differences among them are their objective functions. We theoretically show that if the objective function satisfies some conditions, and the data are sufficiently drawn from independent subspaces, the obtained affinity matrix is always block diagonal. Furthermore, the data sampling can be insufficient if the subspaces are orthogonal. Some existing methods are all special cases. Then we present the Least Squares Regression (LSR) method for subspace segmentation. It takes advantage of data correlation, which is common in real data. LSR encourages a grouping effect which tends to group highly correlated data together. Experimental results on the Hopkins 155 database and Extended Yale Database B show that our method significantly outperforms state-of-the-art methods. Beyond segmentation accuracy, all experiments demonstrate that LSR is much more efficient.

• The paper introduces an LSR-based method that ensures a block diagonal affinity matrix under Enforced Block Diagonal conditions.
• The approach leverages the grouping effect to outperform SSC and LRR, demonstrating lower segmentation error and faster computation on benchmark datasets.
• Experimental results on Hopkins 155 and Extended Yale B validate the method’s robustness against noise and its practical applicability in real-world scenarios.

Robust and Efficient Subspace Segmentation via Least Squares Regression

Subspace segmentation is a key challenge in clustering, particularly relevant to machine learning and computer vision tasks, such as image representation, clustering, and motion segmentation. The paper explores efficient subspace segmentation methods that partition data assumed to originate from multiple linear subspaces. Recent advancements have focused on exploiting sparse and low-rank representations. This paper introduces a novel approach based on Least Squares Regression (LSR) for effective subspace segmentation.

Theoretical Underpinnings

The paper commences with a theoretical exploration of subspace segmentation, presenting conditions under which an optimal solution is guaranteed to result in a block diagonal affinity matrix. These conditions, termed Enforced Block Diagonal (EBD), ensure that data from independent or orthogonal subspaces achieve proper segmentation when sufficiently sampled. The key innovation lies in leveraging LSR, which utilizes the inherent correlation in real-world data to enhance segmentation effectiveness.

LSR is characterized by its ability to group highly correlated data, a feature termed the "grouping effect." This contrasts with Sparse Subspace Clustering (SSC) that can sometimes provide overly sparse solutions, and Low-Rank Representation (LRR), which may lack clarity in its low-rank graph interpretations. Theoretical analysis within the paper demonstrates that LSR is robust against bounded noise, further validating its suitability for real-world datasets.

Numerical Results

Experimental evaluations were carried out on the Hopkins 155 and Extended Yale B databases, establishing LSR's superior performance in comparison to SSC and LRR. On the Hopkins 155 database, LSR demonstrated lower mean segmentation error and reduced computational time. Similarly, LSR achieved higher segmentation accuracy and efficiency on the Extended Yale B database.

Practical and Theoretical Implications

The findings have profound implications both pragmatically and theoretically. Practically, the robust grouping effect of LSR makes it an appealing choice for applications needing reliable segmentation under noisy conditions. Theoretically, the paper's development of EBD conditions enriches understanding of the factors crucial for achieving ideal block diagonal structures in subspace segmentation.

Future Directions

The paper implicitly lays the groundwork for future explorations into refining the LSR model and understanding its interplay with other representation paradigms. Potential avenues include investigating hybrid approaches that integrate different subspace representation methodologies, as well as exploring LSR's utility in emerging fields such as deep learning and high-dimensional data analysis.

In summary, the paper presents a rigorous analysis and compelling evidence for the efficacy of the LSR method in subspace segmentation. The theoretical framework developed not only enriches the current understanding but also provides a robust foundation for future research endeavors in this domain.