[Paper Review] Correlation Adaptive Subspace Segmentation by Trace Lasso

Canyi Lu, Jiashi Feng+2 more|arXiv (Cornell University)|Jan 18, 2015

Sparse and Compressive Sensing Techniques19 references4 citations

TL;DR

This paper proposes Correlation Adaptive Subspace Segmentation (CASS), a novel method that uses trace Lasso to simultaneously enforce sparsity and grouping effects in affinity matrix learning for subspace segmentation. By adaptively balancing ℓ¹ and ℓ² regularization, CASS achieves block-diagonal affinity matrices that outperform SSC, LRR, and LSR on benchmark datasets like Extended Yale B and MNIST, with a 73.80% clustering accuracy on MNIST and 94.03% on 5-subject face clustering.

ABSTRACT

This paper studies the subspace segmentation problem. Given a set of data points drawn from a union of subspaces, the goal is to partition them into their underlying subspaces they were drawn from. The spectral clustering method is used as the framework. It requires to find an affinity matrix which is close to block diagonal, with nonzero entries corresponding to the data point pairs from the same subspace. In this work, we argue that both sparsity and the grouping effect are important for subspace segmentation. A sparse affinity matrix tends to be block diagonal, with less connections between data points from different subspaces. The grouping effect ensures that the highly corrected data which are usually from the same subspace can be grouped together. Sparse Subspace Clustering (SSC), by using $\ell^1$-minimization, encourages sparsity for data selection, but it lacks of the grouping effect. On the contrary, Low-Rank Representation (LRR), by rank minimization, and Least Squares Regression (LSR), by $\ell^2$-regularization, exhibit strong grouping effect, but they are short in subset selection. Thus the obtained affinity matrix is usually very sparse by SSC, yet very dense by LRR and LSR. In this work, we propose the Correlation Adaptive Subspace Segmentation (CASS) method by using trace Lasso. CASS is a data correlation dependent method which simultaneously performs automatic data selection and groups correlated data together. It can be regarded as a method which adaptively balances SSC and LSR. Both theoretical and experimental results show the effectiveness of CASS.

Motivation & Objective

• To address the limitations of existing subspace segmentation methods that lack both sparsity and grouping effects simultaneously.
• To develop a method that adaptively balances sparse representation and group selection based on data correlation.
• To improve affinity matrix construction for spectral clustering by promoting block-diagonal structure.
• To achieve better segmentation accuracy than SSC, LRR, and LSR on real-world data with complex subspace structures.
• To establish theoretical and empirical support for the grouping effect of trace Lasso in subspace learning.

Proposed method

• CASS employs trace Lasso regularization, which combines ℓ¹-norm for sparsity and ℓ²-norm for grouping, to learn an affinity matrix.
• The method solves a convex optimization problem that minimizes a trace Lasso penalty on the representation matrix, promoting both sparsity and correlation-based grouping.
• The optimization is performed via an alternating direction method of multipliers (ADMM) framework to ensure convergence and scalability.
• The affinity matrix derived from the solution is approximately block-diagonal, reflecting true subspace memberships.
• The method is designed to be correlation-adaptive, automatically adjusting the balance between sparsity and grouping based on data structure.
• The approach is applied within a spectral clustering framework, using the learned affinity matrix for final segmentation.

Experimental results

Research questions

• RQ1 Can a method that simultaneously enforces sparsity and grouping effects improve subspace segmentation performance?
• RQ2 How does trace Lasso regularization balance ℓ¹ and ℓ² penalties to enhance affinity matrix structure?
• RQ3 Does the proposed CASS method achieve better segmentation accuracy than SSC, LRR, and LSR on real-world datasets?
• RQ4 Is the grouping effect of trace Lasso theoretically justifiable and empirically observable in subspace clustering?
• RQ5 Can CASS be effectively applied in semi-supervised learning settings with limited labeled data?

Key findings

• On the Extended Yale B database, CASS achieved 94.03% accuracy for 5-subject face clustering, outperforming LSR and other baselines.
• For 8-subject and 10-subject clustering, CASS achieved 91.41% and 81.88% accuracy, respectively, showing significant improvement over SSC and LRR.
• On the MNIST database, CASS achieved 73.80% clustering accuracy, the highest among all compared methods, despite the data's imperfect subspace fit.
• The affinity matrix learned by CASS was visually and quantitatively closer to block-diagonal than those from SSC, LRR, or LSR, confirming improved structure learning.
• In semi-supervised learning, CASS achieved classification accuracies above 90% across all training data settings, demonstrating robustness and utility beyond clustering.
• Theoretical analysis confirms that CASS can recover the true segmentation when subspaces are independent, validating its consistency.