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.
Subspace learning approaches are built on the assumption that all the views are generated from a latent subspace. Its goal is to capture shared latent subspace first and then conduct clustering. The representative methods in this stream are proposed in @cite_26 @cite_5 , which applied canonical correlation analysis (CCA) and kernel CCA to project the multi-view high-dimensional data onto a low-dimensional subspace, respectively. By including robust losses to replace the squared loss used in CCA, @cite_46 provided a convex reformulation of multi-view subspace learning that enforces conditional independence between views. Inspired by deep representation, @cite_35 proposed a DNN-based model combining CCA and autoencoder-based terms to exploit the deep information from two views. Since those CCA based methods are limited by capability of only handling two-view features, tensor CCA @cite_52 generalized CCA to handle the data of an arbitrary number of views by analyzing the covariance tensor of different views.
@cite_35 @cite_26 @cite_52 @cite_5 @cite_46
2158703881 1672347394 2949447882 2056935845
Subspace learning seeks a low dimensional representation of data that enables accurate reconstruction. However, in many applications, data is obtained from multiple sources rather than a single source (e.g. an object might be viewed by cameras at different angles, or a document might consist of text and images). The conditional independence of separate sources imposes constraints on their shared latent representation, which, if respected, can improve the quality of a learned low dimensional representation. In this paper, we present a convex formulation of multi-view subspace learning that enforces conditional independence while reducing dimensionality. For this formulation, we develop an efficient algorithm that recovers an optimal data reconstruction by exploiting an implicit convex regularizer, then recovers the corresponding latent representation and reconstruction model, jointly and optimally. Experiments illustrate that the proposed method produces high quality results. Canonical correlation analysis (CCA) has proven an effective tool for two-view dimension reduction due to its profound theoretical foundation and success in practical applications. In respect of multi-view learning, however, it is limited by its capability of only handling data represented by two-view features, while in many real-world applications, the number of views is frequently many more. Although the ad hoc way of simultaneously exploring all possible pairs of features can numerically deal with multi-view data, it ignores the high order statistics (correlation information) which can only be discovered by simultaneously exploring all features. Therefore, in this work, we develop tensor CCA (TCCA) which straightforwardly yet naturally generalizes CCA to handle the data of an arbitrary number of views by analyzing the covariance tensor of the different views. TCCA aims to directly maximize the canonical correlation of multiple (more than two) views. Crucially, we prove that the main problem of multi-view canonical correlation maximization is equivalent to finding the best rank- @math approximation of the data covariance tensor, which can be solved efficiently using the well-known alternating least squares (ALS) algorithm. As a consequence, the high order correlation information contained in the different views is explored and thus a more reliable common subspace shared by all features can be obtained. In addition, a non-linear extension of TCCA is presented. Experiments on various challenge tasks, including large scale biometric structure prediction, internet advertisement classification, and web image annotation, demonstrate the effectiveness of the proposed method 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. To uncover an appropriate latent subspace for data representation, in this paper we propose a novel Robust Structured Subspace Learning (RSSL) algorithm by integrating image understanding and feature learning into a joint learning framework. The learned subspace is adopted as an intermediate space to reduce the semantic gap between the low-level visual features and the high-level semantics. To guarantee the subspace to be compact and discriminative, the intrinsic geometric structure of data, and the local and global structural consistencies over labels are exploited simultaneously in the proposed algorithm. Besides, we adopt the @math -norm for the formulations of loss function and regularization respectively to make our algorithm robust to the outliers and noise. An efficient algorithm is designed to solve the proposed optimization problem. It is noted that the proposed framework is a general one which can leverage several well-known algorithms as special cases and elucidate their intrinsic relationships. To validate the effectiveness of the proposed method, extensive experiments are conducted on diversity datasets for different image understanding tasks, i.e., image tagging, clustering, and classification, and the more encouraging results are achieved compared with some state-of-the-art approaches.