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.

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.

We consider deep neural networks (DNNs) with a Lipschitz continuous activation function and with weight matrices of variable widths. We establish a uniform convergence analysis framework in which sufficient conditions on weight matrices and bias vectors together with the Lipschitz constant are provided to ensure uniform convergence of DNNs to a meaningful function as the number of their layers tends to infinity. In the framework, special results on uniform convergence of DNNs with a fixed width, bounded widths and unbounded widths are presented. In particular, as convolutional neural networks are special DNNs with weight matrices of increasing widths, we put forward conditions on the mask sequence which lead to uniform convergence of the resulting convolutional neural networks. The Lipschitz continuity assumption on the activation functions allows us to include in our theory most of commonly used activation functions in applications.

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.

Kernel methods are built upon the mathematical theory of reproducing kernels and reproducing kernel Hilbert spaces. They enjoy good interpretability thanks to the solid mathematical foundation. Recently, motivated by deep neural networks in deep learning, which construct learning functions by successive compositions of activation functions and linear functions, a class of methods termed as deep kernel learning has appeared in the literature. The core of deep kernel learning is hierarchical kernels that are constructed from a base reproducing kernel by successive compositions. In this paper, we characterize the corresponding reproducing kernel Hilbert spaces of hierarchical kernels, and study conditions ensuring that the reproducing kernel Hilbert space will be expanding as the layer of hierarchical kernels increases. The results will answer whether the expressive power of hierarchical kernels will be improving as the layer increases, and give guidance to the construction of hierarchical kernels for deep kernel learning.

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.

A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error, and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we illustrate how reproducing kernel Banach spaces with the ℓ1norm can be applied to improve the learning rate estimate of ℓ1-regularization in machine learning.

Reproducing kernel Hilbert space (RKHS) methods have become powerful tools in machine learning. However, their kernels, which measure similarity of inputs, are required to be symmetric, constraining certain applications in practice. Furthermore, the celebrated representer theorem only applies to regularizers induced by the norm of an RKHS. To remove these limitations, we introduce the notion of reproducing kernel Banach spaces (RKBS) for pairs of reflexive Banach spaces of functions by making use of semi-inner-products and the duality mapping. As applications, we develop the framework of RKBS standard learning schemes including minimal norm interpolation, regularization network, and support vector machines. In particular, existence, uniqueness and representer theorems are established.

In this paper we investigate conditions on the features of a continuous kernel so that it may approximate an arbitrary continuous target function uniformly on any compact subset of the input space. A number of concrete examples are given of kernels with this universal approximating property.