Tensorized Multi-view Clustering via Hyper-graph Regularization

In this paper, we address the multiview nonlinear subspace representation problem. Traditional multiview subspace learning methods assume that the heterogeneous features of the data usually lie within the union of multiple linear subspaces. However, instead of linear subspaces, the data feature actually resides in multiple nonlinear subspaces in many real-world applications, resulting in unsatisfactory clustering performance. To overcome this, we propose a hyper-Laplacian regularized multilinear multiview self-representation model, which is referred to as HLR-M2VS, to jointly learn multiple views correlation and a local geometrical structure in a unified tensor space and view-specific self-representation feature spaces, respectively. In unified tensor space, a well-founded tensor low-rank regularization is adopted to impose on the self-representation coefficient tensor to ensure global consensus among different views. In view-specific feature space, hypergraph-induced hyper-Laplacian regularization is utilized to preserve the local geometrical structure embedded in a high-dimensional ambient space. An efficient algorithm is then derived to solve the optimization problem of the established model with theoretical convergence guarantee. Furthermore, the proposed model can be extended to semisupervised classification without introducing any additional parameters. An extensive experiment of our method is conducted on many challenging datasets, where a clear advance over state-of-the-art multiview clustering and multiview semisupervised classification approaches is achieved.

Despite satisfactory clustering performance, current subspace-based multi-view clustering methods still suffer from the following limitations. 1) They usually concentrate on the data features in linear subspaces and ignore the features in nonlinear subspaces. 2) They treat all singular values equally without considering their different contribution degrees, leading to suboptimal problems. Based on the above considerations, we propose tensorial multi-linear multi-view clustering via the weighted Schatten-p norm, named TM2vC. TM2vC integrates high-order relationships learning and latent structure learning of multi-view data into one framework. We apply third-order tensors stacked by low-dimensional representations to capture high-order relationships among multivariate data and the weighted Schatten-p norm to distinguish different singular values. Additionally, we employ hypergraph constraints to conserve high-order local geometric structures in the high-dimensional subspace. Comprehensive experiments on diverse datasets verify the effectiveness and superiority of the proposed TM2vC on six indicators.

Asymmetric double-winged multi-view clustering network for exploring diverse and consistent information

Multi-view clustering via spectral partitioning and local refinement

Deep multi-view subspace clustering has achieved promising performance compared with other multi-view clustering. However, existing deep multi-view subspace clustering only considers the global structure for all views, and they ignore the local geometric structure among each view. In addition, they cannot learn discriminative feature on different clusters of different views, i.e., inter-cluster difference. To solve these problems, in this paper, we propose a novel Deep Multi-view Subspace Clustering with Unified and Discriminative Learning (DMSC-UDL). DMSC-UDL combines global and local structures with self-expression layer. The global and local structures help each other forward and achieve small distance between samples of the same cluster. To make samples in different clusters of different views farther, DMSC-UDL uses a discriminative constraint between different views. In this way, DMSC-UDL makes the same cluster's samples have large weights, while different clusters’ samples have small weights. Thus, it can learn a better shared connection matrix for multi-view clustering. Extensive experimental results reveal that the proposed multi-view clustering method is superior to several state-of-the-art multi-view clustering methods in terms of performance.

Multi-view clustering has become a widely studied problem in the area of unsupervised learning. It aims to integrate multiple views by taking advantages of the consensus and complimentary information from multiple views. Most of the existing works in multi-view clustering utilize the vector-based representation for features in each view. However, in many real-world applications, instances are represented by graphs, where those vector-based models cannot fully capture the structure of the graphs from each view. To solve this problem, in this paper we propose a Multi-view Clustering framework on graph instances with Graph Embedding (MCGE). Specifically, we model the multi-view graph data as tensors and apply tensor factorization to learn the multi-view graph embeddings, thereby capturing the local structure of graphs. We build an iterative framework by incorporating multi-view graph embedding into the multi-view clustering task on graph instances, jointly performing multi-view clustering and multi-view graph embedding simultaneously. The multi-view clustering results are used for refining the multi-view graph embedding, and the updated multi-view graph embedding results further improve the multi-view clustering. Extensive experiments on two real brain network datasets (i.e., HIV and Bipolar) demonstrate the superior performance of the proposed MCGE approach in multi-view connectome analysis for clinical investigation and application.

This paper presents a robust reliability analysis method for systems of multimodular redundant (MMR) controllers using the method of partitioning and parallel processing of a Markov chain (PPMC). A Markov chain is formulated to represent the N distinct states of the MMR controllers. Such a Markov chain has N2 directed edges, and each edge corresponds to a transition probability between a pair of start and end states. Because N can be easily increased substantially, the system reliability analysis may require large computational resources, such as the central processing unit usage and memory occupation. By the PPMC, a Markov chain's transition probability matrix can be partitioned and reordered, such that the system reliability can be evaluated through only the diagonal submatrices of the transition probability matrix. In addition, calculations regarding the submatrices are independent of each other and thus can be conducted in parallel to assure the efficiency. The simulation results show that, compared with the sequential method applied to an intact Markov chain, the proposed PPMC can improve the performance and produce allowable accuracy for the reliability analysis on large-scale systems of MMR controllers.

Multi-view clustering aims to capture the multiple views inherent information by identifying the data clustering that reflects distinct features of datasets. Since there is a consensus in literature that different views of a dataset share a common latent structure, most existing multi-view subspace learning methods rely on the nuclear norm to seek the low-rank representation of the underlying subspace. However, the nuclear norm often fails to distinguish the variance of features for each cluster due to its convex nature and data tends to fall in multiple non-linear subspaces for multi-dimensional datasets. To address these problems, we propose a new and novel multi-view clustering method (HL-L21-TLD-MSC) that unifies the Hyper-Laplacian (HL) and exclusive ℓ 2,1 (L21) regularization with the Tensor Log-Determinant Rank Minimization (TLD) setting. Specifically, the hyper-Laplacian regularization maintains the local geometrical structure that makes the estimation prune to nonlinearities, and the mixed ℓ 2,1 and ℓ 1,2 regularization provides the joint sparsity within-cluster as well as the exclusive sparsity between-cluster. Furthermore, a log-determinant function is used as a tighter tensor rank approximation to discriminate the dimension of features. An efficient alternating algorithm is then derived to optimize the proposed model, and the construction of a convergent sequence to the Karush-Kuhn-Tucker (KKT) critical point solution is mathematically validated in detail. Extensive experiments are conducted on ten well-known datasets to demonstrate that the proposed approach outperforms the existing state-of-the-art approaches with various scenarios, in which, six of them achieve perfect results under our framework developed in this article, demonstrating highl effectiveness for the proposed approach.

Some fields of an encoded speech or audio signal's bit stream, which varies according to encoder's type, can be modeled by Markov chains. In this paper, a novel “syndrome code data hiding using Markov chains” is proposed where secret data is embedded into the syndromes of the C(N, K) linear block codes as in matrix embedding. However, the proposed method randomly chooses a codeword from 2K possible code words according to the Markov chain's transition probabilities, which is different from matrix embedding method. Performance of the proposed method is compared with that of least significant bit and matrix embedding methods employed on GSM 6.10 coder. The simulation results show that data hiding using statistical modeling with Markov chains preserves the original bit stream's entropy, leading to undetectability in terms of steganalysis. Unfortunately secret data embedding efficiency is decreased.

In this paper, we study some strong limit theorems relative to the geometric average of random transition probabilities of nonhomogeneous Markov chains indexed by a tree. This result is an extension of some strong limit theorems relative to the geometric average of random transition probabilities of arbitrary finite nonhomogeneous Markov chains.

Multi-view clustering with adaptive anchor and bipartite graph learning

Graph-based multi-view clustering can effectively reveal the latent cluster structure of multi-view data, however, it remains challenging to construct high-quality graphs by exploring the discriminant information from data. Most of the existing graph-based methods usually construct the graph in the original space or subspace, which may not take the differences of semantic information in different spaces into considerations. Moreover, existing approaches generally measure the relevance of the pair-wise views, which may result in insufficient information exploitation and limited graph representation. To remedy these issues, we proposed an innovative multi-view clustering algorithm called reciprocal consistency graph learning (RCGL), which learns consistency representation by aligning the semantic information from three spaces simultaneously to uncover latent complementarity and consistency among different views. Specifically, we construct the low-dimensional space based on linear projection to preserve essential semantic information. Furthermore, RCGL maps the low-dimension representation from linear space into nonlinear space via Hilbert–Schmidt independence criterion, to explore the nonlinear information hidden in data. Moreover, multiple graphs in kernel space are extended to tensor space for learning a high-order graph, which is helpful to better measure the correlation between multiple views. Finally, the objective function is solved by an alternating optimization scheme. Extensive experiments on five benchmark multi-view datasets demonstrate that RCGL outperforms state-of-the-art baselines markedly.

Hyper-Laplacian regularized multi-view subspace clustering with low-rank tensor constraint

In the era of big data, data may come from multiple sources, known as multi-view data. Multi-view clustering aims at generating better clusters by exploiting complementary and consistent information from multiple views rather than relying on the individual view. Due to inevitable system errors caused by data-captured sensors or others, the data in each view may be erroneous. Various types of errors behave differently and inconsistently in each view. More precisely, error could exhibit as noise and corruptions in reality. Unfortunately, none of the existing multi-view clustering approaches handle all of these error types. Consequently, their clustering performance is dramatically degraded. In this paper, we propose a novel Markov chain method for Error-Robust Multi-View Clustering (EMVC). By decomposing each view into a shared transition probability matrix and error matrix and imposing structured sparsity-inducing norms on error matrices, we characterize and handle typical types of errors explicitly. To solve the challenging optimization problem, we propose a new efficient algorithm based on Augmented Lagrangian Multipliers and prove its convergence rigorously. Experimental results on various synthetic and real-world datasets show the superiority of the proposed EMVC method over the baseline methods and its robustness against different types of errors.

The task of planning work while the implementation of the project, where you can separate several stages, is discussed in the article. Dependent stages are allowed, the fulfillment of which is possible only after the successful completion of a particular stage or group of stages. The approach for receiving the maximum mean income under planning for the period under review is offered, positive or negative income is associated with the fulfillment or non-fulfillment of certain stages of the project. The basis of approach and Markov chain theory are the method of dynamic programming. All the variables of the given problem are described, the necessary formulas and equations for numerical calculations based on the recursive Bellman relation are given. The processing of expert estimates (obtaining the transition probabilities of Markov chains) is proposed to be made using the methods of Saaty. The decision result is the selection of admissible managements. Managements (management decisions) allow to adapt with various scenarios which may occur on the implementation stages of projects.