Dual Information Enhanced Multiview Attributed Graph Clustering.

Multiview attributed graph clustering is an important approach to partition multiview data based on the attribute characteristics and adjacent matrices from different views. Some attempts have been made in using graph neural network (GNN), which have achieved promising clustering performance. Despite this, few of them pay attention to the inherent specific information embedded in multiple views. Meanwhile, they are incapable of recovering the latent high-level representation from the low-level ones, greatly limiting the downstream clustering performance. To fill these gaps, a novel dual information enhanced multiview attributed graph clustering (DIAGC) method is proposed in this article. Specifically, the proposed method introduces the specific information reconstruction (SIR) module to disentangle the explorations of the consensus and specific information from multiple views, which enables graph convolutional network (GCN) to capture the more essential low-level representations. Besides, the contrastive learning (CL) module maximizes the agreement between the latent high-level representation and low-level ones and enables the high-level representation to satisfy the desired clustering structure with the help of the self-supervised clustering (SC) module. Extensive experiments on several real-world benchmarks demonstrate the effectiveness of the proposed DIAGC method compared with the state-of-the-art baselines.

Incomplete Data Meets Uncoupled Case: A Challenging Task of Multiview Clustering.

Incomplete multiview clustering (IMC) methods have achieved remarkable progress by exploring the complementary information and consensus representation of incomplete multiview data. However, to our best knowledge, none of the existing methods attempts to handle the uncoupled and incomplete data simultaneously, which affects their generalization ability in real-world scenarios. For uncoupled incomplete data, the unclear and partial cross-view correlation introduces the difficulty to explore the complementary information between views, which results in the unpromising clustering performance for the existing multiview clustering methods. Besides, the presence of hyperparameters limits their applications. To fill these gaps, a novel uncoupled IMC (UIMC) method is proposed in this article. Specifically, UIMC develops a joint framework for feature inferring and recoupling. The high-order correlations of all views are explored by performing a tensor singular value decomposition (t-SVD)-based tensor nuclear norm (TNN) on recoupled and inferred self-representation matrices. Moreover, all hyperparameters of the UIMC method are updated in an exploratory manner. Extensive experiments on six widely used real-world datasets have confirmed the superiority of the proposed method in handling the uncoupled incomplete multiview data compared with the state-of-the-art methods.

A Tensor Approach for Uncoupled Multiview Clustering.

Multiview clustering plays an important part in unsupervised learning. Although the existing methods have shown promising clustering performances, most of them assume that the data is completely coupled between different views, which is unfortunately not always ensured in real-world applications. The clustering performance of these methods drops dramatically when handling the uncoupled data. The main reason is that: 1) cross-view correlation of uncoupled data is unclear, which limits the existing multiview clustering methods to explore the complementary information between views and 2) features from different views are uncoupled with each other, which may mislead the multiview clustering methods to partition data into wrong clusters. To address these limitations, we propose a tensor approach for uncoupled multiview clustering (T-UMC) in this article. Instead of pairwise correlation, T-UMC chooses a most reliable view by view-specific silhouette coefficient (VSSC) at first, and then couples the self-representation matrix of each view with it by pairwise cross-view coupling learning. After that, by integrating recoupled self-representation matrices into a third-order tensor, the high-order correlations of all views are explored with tensor singular value decomposition (t-SVD)-based tensor nuclear norm (TNN). And the view-specific local structures of each individual view are also preserved with the local structure learning scheme with manifold learning. Besides, the physical meaning of view-specific coupling matrix is also discussed in this article. Extensive experiments on six commonly used benchmark datasets have demonstrated the superiority of the proposed method compared with the state-of-the-art multiview clustering methods.

Convergence of deep convolutional neural networks.

Convergence of deep neural networks as the depth of the networks tends to infinity is fundamental in building the mathematical foundation for deep learning. In a previous study, we investigated this question for deep networks with the Rectified Linear Unit (ReLU) activation function and with a fixed width. This does not cover the important convolutional neural networks where the widths are increased from layer to layer. For this reason, we first study convergence of general ReLU networks with increased widths and then apply the results obtained to deep convolutional neural networks. It turns out the convergence reduces to convergence of infinite products of matrices with increased sizes, which has not been considered in the literature. We establish sufficient conditions for convergence of such infinite products of matrices. Based on the conditions, we present sufficient conditions for pointwise convergence of general deep ReLU networks with increasing widths, and as well as pointwise convergence of deep ReLU convolutional neural networks.

Margin Error Bounds for Support Vector Machines on Reproducing Kernel Banach Spaces.

Support vector machines, which maximize the margin from patterns to the separation hyperplane subject to correct classification, have received remarkable success in machine learning. Margin error bounds based on Hilbert spaces have been introduced in the literature to justify the strategy of maximizing the margin in SVM. Recently, there has been much interest in developing Banach space methods for machine learning. Large margin classification in Banach spaces is a focus of such attempts. In this letter we establish a margin error bound for the SVM on reproducing kernel Banach spaces, thus supplying statistical justification for large-margin classification in Banach spaces.