On Unifying Multi-View Self-Representations for Clustering by Tensor Multi-Rank Minimization (1610.07126v3)

Abstract: In this paper, we address the multi-view subspace clustering problem. Our method utilizes the circulant algebra for tensor, which is constructed by stacking the subspace representation matrices of different views and then rotating, to capture the low rank tensor subspace so that the refinement of the view-specific subspaces can be achieved, as well as the high order correlations underlying multi-view data can be explored.} By introducing a recently proposed tensor factorization, namely tensor-Singular Value Decomposition (t-SVD) \cite{kilmer13}, we can impose a new type of low-rank tensor constraint on the rotated tensor to capture the complementary information from multiple views. Different from traditional unfolding based tensor norm, this low-rank tensor constraint has optimality properties similar to that of matrix rank derived from SVD, so the complementary information among views can be explored more efficiently and thoroughly. The established model, called t-SVD based Multi-view Subspace Clustering (t-SVD-MSC), falls into the applicable scope of augmented Lagrangian method, and its minimization problem can be efficiently solved with theoretical convergence guarantee and relatively low computational complexity. Extensive experimental testing on eight challenging image dataset shows that the proposed method has achieved highly competent objective performance compared to several state-of-the-art multi-view clustering methods.

• The paper presents a novel t-SVD based multi-view clustering method that unifies subspace representations with low-rank tensor modeling.
• It leverages tensor nuclear norm minimization and Fourier domain rotation to capture high-order correlations while reducing computational complexity.
• Empirical evaluations on image datasets demonstrate superior clustering performance compared to state-of-the-art methods.

Insights on "On Unifying Multi-View Self-Representations for Clustering by Tensor Multi-Rank Minimization"

The paper addresses the intricate problem of multi-view subspace clustering, aiming to integrate representation matrices from multiple views into a unified model using tensor algebra. The authors propose a novel approach termed as t-SVD based Multi-view Subspace Clustering (t-SVD-MSC). This method capitalizes on the concept of tensor-Singular Value Decomposition (t-SVD), a recent development in tensor factorization that effectively manages high-dimensional data structures by imposing low-rank tensor constraints.

Methodological Contribution

The core contribution lies in the utilization of t-SVD to construct a low-rank tensor subspace by rotating the stacked subspace representation matrices of different views. The motivation behind introducing a t-SVD-based model is to overcome limitations associated with traditional unfolding methods which lack a clear physical interpretation for tensors. The t-SVD provides clear optimality properties similar to matrix rank derivations, offering an optimal approximation and bringing forth the tensor nuclear norm (t-TNN) as its tightest convex relaxation.

This rigorous tensor approach allows the model to explore and propagate complementary information among multiple views, ensuring consensus and capturing high-order correlations in multi-view data. By appropriately rotating the tensor, the self-representation coefficients are preserved in the Fourier domain, which is shown to better manage computational complexity and maintain efficient convergence during optimization.

Optimization and Algorithmic Framework

The problem is structured into an optimization framework solvable through the augmented Lagrangian method, providing theoretical convergence assurance. The paper meticulously designs an algorithm that iteratively updates variables associated with subspace representations and auxiliary tensors while minimizing computational burden. The algorithm's integrity is secured through adaptive parameter updates and the efficient resolution of tensor multi-rank minimization subproblems.

Empirical Evaluation

The method is subjected to extensive experimental evaluation across multiple challenging image datasets spanning face, scene, and generic object clustering. Comparative analysis against state-of-the-art multi-view clustering techniques demonstrates superior performance of t-SVD-MSC, especially when leveraging CNN-based features from modern deep networks. Tangibly, the approach exhibits robust clustering performance, outperforming traditional tensor decomposition methods and strategies relying solely on singular feature perspectives.

Implications and Future Directions

Practically, the implications of this research suggest substantial advancements in the field of multi-view learning and tensor-based data analysis. The approach can be directly applied to tasks requiring nuanced integration of diverse data sources or features, thus broadening its utility in computer vision, bioinformatics, and multimedia data processing.

Theoretically, the work enriches existing paradigms of tensor decompositions with novel insights into aligning data representations across multiple domains. Future research could delve into adapting such tensor-based methodologies to evolving machine learning architectures and exploring their applicability in real-time data environments.

In conclusion, the paper presents a well-founded, computationally efficient framework for multi-view clustering, advancing the dialogue in utilizing higher-order tensor representations for comprehensive data analysis. The methodological rigor and positive empirical results underscore its importance as a substantial contribution to the multi-view learning landscape.