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) 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.
Besides CCA, the recent proposed subspace clustering methods @cite_28 @cite_25 resorted to explore the relationship between samples with self-representation ( e.g., sparse subspace clustering (SSC) @cite_23 and low-rank representation (LRR) @cite_21 ) in multi-view setting. Our approach is closely related to @cite_25 , which extended the LRR based subspace clustering to multi-view by employing the rank-sum of different mode unfoldings to constrain the subspace coefficient tensor. However, such a kind of tensor constraint lacks a clear physical meaning for general tensor, so that it can not thoroughly explore the complementary information among different views. On the contrary, the high order constraint within our model is built upon a new tensor decomposition scheme @cite_16 @cite_2 , which is referred to as t-SVD and has been applied to various tasks, such as image reconstruction and tensor completion @cite_53 @cite_41 @cite_20 . Therefore, the proposed model possesses good theoretical properties and clear physical meaning for handling the subspace representation tensor. The detailed motivation will be presented in Section .
@cite_28 @cite_41 @cite_53 @cite_21 @cite_23 @cite_2 @cite_16 @cite_25 @cite_20
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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) 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. 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 l1-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 l2-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. In this paper, we explore the problem of multiview subspace clustering. We introduce a low-rank tensor constraint to explore the complementary information from multiple views and, accordingly, establish a novel method called Low-rank Tensor constrained Multiview Subspace Clustering (LT-MSC). Our method regards the subspace representation matrices of different views as a tensor, which captures dexterously the high order correlations underlying multiview data. Then the tensor is equipped with a low-rank constraint, which models elegantly the cross information among different views, reduces effectually the redundancy of the learned subspace representations, and improves the accuracy of clustering as well. The inference process of the affinity matrix for clustering is formulated as a tensor nuclear norm minimization problem, constrained with an additional L2,1-norm regularizer and some linear equalities. The minimization problem is convex and thus can be solved efficiently by an Augmented Lagrangian Alternating Direction Minimization (AL-ADM) method. Extensive experimental results on four benchmark datasets show the effectiveness of our proposed LT-MSC method. An important problem in analyzing big data is subspace clustering , i.e., to represent a collection of points in a high-dimensional space via the union of low-dimensional subspaces. Sparse subspace clustering (SSC) and Low-rank representation (LRR) are the state-of-the-art methods for this task. These two methods are fundamentally similar in that both are based on convex optimization exploiting the intuition of “Self-Expressiveness”. The main difference is that the SSC minimizes the vector @math norm of the representation matrix to induce sparsity while LRR minimizes the nuclear norm (aka trace norm) to promote a low-rank structure. Because the representation matrix is often simultaneously sparse and low-rank, we propose a new algorithm, termed Low-rank sparse subspace clustering (LRSSC), by the combining SSC and LRR, and develop theoretical guarantees of the success of the algorithm. The results reveal interesting insights into the strengths and the weaknesses of SSC and LRR, and demonstrate how the LRSSC can take advantage of both methods in preserving the “Self-Expressiveness Property” and “Graph Connectivity” at the same time. A byproduct of our analysis is that it also expands the theoretical guarantee of SSC to handle cases when the subspaces have arbitrarily small canonical angles but are “nearly independent”.