Constructing the L2-Graph for Robust Subspace Learning and Subspace Clustering (1209.0841v7)

Abstract: Under the framework of graph-based learning, the key to robust subspace clustering and subspace learning is to obtain a good similarity graph that eliminates the effects of errors and retains only connections between the data points from the same subspace (i.e., intra-subspace data points). Recent works achieve good performance by modeling errors into their objective functions to remove the errors from the inputs. However, these approaches face the limitations that the structure of errors should be known prior and a complex convex problem must be solved. In this paper, we present a novel method to eliminate the effects of the errors from the projection space (representation) rather than from the input space. We first prove that $\ell_1$-, $\ell_2$-, $\ell_{\infty}$-, and nuclear-norm based linear projection spaces share the property of Intra-subspace Projection Dominance (IPD), i.e., the coefficients over intra-subspace data points are larger than those over inter-subspace data points. Based on this property, we introduce a method to construct a sparse similarity graph, called L2-Graph. The subspace clustering and subspace learning algorithms are developed upon L2-Graph. Experiments show that L2-Graph algorithms outperform the state-of-the-art methods for feature extraction, image clustering, and motion segmentation in terms of accuracy, robustness, and time efficiency.

• The paper introduces the Intra-subspace Projection Dominance (IPD) principle to construct the L2-Graph, effectively eliminating errors in projection spaces for robust subspace clustering.
• It employs a hard thresholding strategy on ℓ2-norm projections, bypassing complex convex optimization and significantly enhancing computational efficiency.
• Experimental results on image datasets demonstrate that L2-Graph outperforms state-of-the-art methods in clustering accuracy and noise robustness, highlighting its applicability in high-dimensional data analysis.

Overview of the L2-Graph Method for Robust Subspace Learning and Clustering

The paper presents a novel approach to subspace clustering and learning under graph-based frameworks by introducing the L2-Graph method. Its core contribution lies in harnessing an observable property of linear projection spaces termed Intra-subspace Projection Dominance (IPD). IPD suggests that coefficients corresponding to data points from the same subspace (intra-subspace) exhibit a dominance over coefficients from different subspaces (inter-subspace). This property is leveraged to construct sparse similarity graphs without requiring a priori error structure information.

Methodological Contribution

In the proposed L2-Graph, instead of modeling the data space errors within objective functions, the method focuses on eliminating errors directly from the projection space. This contrasts with existing methodologies that typically require substantial computational effort to handle noise using convex optimization problems, which presuppose knowledge of the error structure.

1. Intra-subspace Projection Dominance (IPD): The paper establishes this foundation across $\ell_1$-, $\ell_2$-, $\ell_\infty$-, and nuclear-norm based projections. This is theoretically supported, suggesting that trivial coefficients in the projection space tend to represent inter-subspace relationships and errors rather than intra-subspace data points.
2. Construction of L2-Graph: By utilizing IPD, the L2-Graph is constructed using $\ell_2$-norm projections of the dataset. A hard thresholding is applied where coefficients below a certain level, suggestive of errors, are eliminated. The resultant graph maintains connections primarily between data points of the same subspace, aiding in robust feature extraction and clustering.
3. Algorithm Efficiency: L2-Graph exhibits a computational advantage due to its analytical solution, bypassing the necessity for complex convex optimization associated with other methods, such as Sparse Subspace Clustering (SSC) and Low Rank Representation (LRR).

Empirical Validation and Performance

The paper rigorously tests the L2-Graph across multiple well-known image datasets, including ExYaleB, AR, and multiple sessions from MPIE, in the contexts of both subspace clustering and subspace learning. Experiments extend to scenarios involving significant noise, real-world occlusions, and even motion segmentation (Hopkins155 database). The results are strongly indicative of superior performance by L2-Graph compared to state-of-the-art methods concerning clustering accuracy and robustness against noise.

Theoretical Implications and Future Directions

The theoretical groundwork laid for IPD in $\ell_p$-norm and nuclear-norm settings introduces intriguing avenues for further exploration. This could include deeper analyses on parameter sensitivity, embedding principles for automatic subspace dimensionality selection, and adaptation to non-linear manifold learning scenarios.

Practically speaking, the L2-Graph's error-tolerant attributes position it as a promising tool in applications involving highly corrupted or incomplete datasets. Such datasets are prevalent in practical scenarios, including video processing, biomedical imaging, and high-dimensional sensor data analytics.

In conclusion, the L2-Graph offers a significant contribution to learning and clustering frameworks by innovatively addressing error handling in subspace projections. Its efficiency and robustness across diverse noise and data conditions hold substantial potential for broader applications in AI and data science, specifically within environments characterized by complex, high-dimensional data distributions.