Incomplete multi-view clustering (IMVC) aims to reveal shared clustering structures within multi-view data, where only partial views of the samples are available. Existing IMVC methods primarily suffer from two issues: 1) Imputation-based methods inevitably introduce inaccurate imputations, which in turn degrade clustering performance; 2) Imputation-free methods are susceptible to unbalanced information among views and fail to fully exploit shared information. To address these issues, we propose a novel method based on variational autoencoders. Specifically, we adopt multiple view-specific encoders to extract information from each view and utilize the Product-of-Experts approach to efficiently aggregate information to obtain the common representation. To enhance the shared information in the common representation, we introduce a coherence objective to mitigate the influence of information imbalance. By incorporating the Mixture-of-Gaussians prior information into the latent representation, our proposed method is able to learn the common representation with clustering-friendly structures. Extensive experiments on four datasets show that our method achieves competitive clustering performance compared with state-of-the-art methods.

The difficulty of partial multi-view multi-label learning lies in coupling the consensus of multi-view data with the task relevance of multi-label classification, under the condition where partial views and labels are unavailable. In this paper, we seek to compress cross-view representation to maximize the proportion of shared information to better predict semantic tags. To achieve this, we establish a model consistent with the information bottleneck theory for learning cross-view shared representation, minimizing non-shared information while maintaining feature validity to help increase the purity of task-relevant information. Furthermore, we model multi-label prototype instances in the latent space and learn label correlations in a data-driven manner. Our method outperforms existing state-of-the-art methods on multiple public datasets while exhibiting good compatibility with both partial and complete data. Finally, we experimentally reveal the importance of condensing shared information under the premise of information balancing, in the process of multi-view information encoding and compression.

Many machine learning problems can be formulated as non-convex multi-player games. Due to non-convexity, it is challenging to obtain the existence condition of the global Nash equilibrium (NE) and design theoretically guaranteed algorithms. This paper studies a class of non-convex multi-player games, where players' payoff functions consist of canonical functions and quadratic operators. We leverage conjugate properties to transform the complementary problem into a variational inequality (VI) problem using a continuous pseudo-gradient mapping. We prove the existence condition of the global NE as the solution to the VI problem satisfies a duality relation. We then design an ordinary differential equation to approach the global NE with an exponential convergence rate. For practical implementation, we derive a discretized algorithm and apply it to two scenarios: multi-player games with generalized monotonicity and multi-player potential games. In the two settings, step sizes are required to be O(1/k) and O(1/√k) to yield the convergence rates of O(1/ k) and O(1/√k), respectively. Extensive experiments on robust neural network training and sensor network localization validate our theory. Our code is available at https://github.com/GuanpuChen/Global-NE.

In recent years, many incomplete multi-view clustering methods have been proposed to address the challenging and new clustering task on incomplete multi-view data whose part of view representations are not fully collected for some samples. Although extensive experiments have validated the effectiveness of these methods for handling the incomplete learning issue, a common issue exists, i.e., these methods all ignore the discriminative/important difference of discriminative features and noisy features. In this paper, to address the above issue, a new incomplete multi-view clustering model, called Graph Regularized and fEature Aware maTrix Factorization (GreatF), is proposed. Different from the existing methods, we introduce an adaptive feature weighting constraint to the matrix factorization-based multi-view representation learning model. With this weighting constraint, the effect of the discriminative features can be enhanced while the negative effect caused by the redundant and noisy features can be eliminated for the model optimization; thus, the robustness of the model can be enhanced. In addition, in this work, we designed a new graph-embedded consensus representation learning term in which consensus representation learning and structure information preservation are integrated into a joint model with one term. In particular, this term provides a more concise approach to obtain the structured consensus representation from incomplete multi-view data. Experimental results on four well-known datasets demonstrate that GreatF performs better than the state-of-the-art incomplete multi-view clustering methods.

There has been significant recent interest in a class of three-player leader-follower game models in many important cybersecurity scenarios. In such a tri-level hierarchical structure, a defender usually serves as a leader, dominating the decision process by the Stackelberg equilibrium (SE) strategy. However, such a leader-follower scheme may not always work, and the Nash equilibrium (NE) strategy may provide an alternative choice. Thus, we need to reveal the consistency between SE and NE in the three-player model to help the leader evaluate its strategy impact and avoid a choice dilemma. To this end, we first provide a necessary and sufficient condition such that each SE is an NE, which not only provides access to seek a satisfactory SE but also makes a criterion for an obtained SE. Then, we apply the results for case studies with a unique SE or with at least one SE being an NE. Moreover, when the consistency condition falls short, we give an upper bound of the deviation between SE and NE to help the leader tolerably adopt an SE strategy. Finally, we apply our consistency analysis to practical scenarios, including secure wireless transmission and advanced persistent threat defense.

Due to the rapid development of multimedia technology and sensor technology, multi-view clustering (MVC) has become a research hotspot in machine learning, data mining, and other fields and has been developed significantly in the past decades. Compared with single-view clustering, MVC improves clustering performance by exploiting complementary and consistent information among different views. Such methods are all based on the assumption of complete views, which means that all the views of all the samples exist. It limits the application of MVC, because there are always missing views in practical situations. In recent years, many methods have been proposed to solve the incomplete MVC (IMVC) problem and a kind of popular method is based on matrix factorization (MF). However, such methods generally cannot deal with new samples and do not take into account the imbalance of information between different views. To address these two issues, we propose a new IMVC method, in which a novel and simple graph regularized projective consensus representation learning model is formulated for incomplete multi-view data clustering task. Compared with the existing methods, our method not only can obtain a set of projections to handle new samples but also can explore information of multiple views in a balanced way by learning the consensus representation in a unified low-dimensional subspace. In addition, a graph constraint is imposed on the consensus representation to mine the structural information inside the data. Experimental results on four datasets show that our method successfully accomplishes the IMVC task and obtain the best clustering performance most of the time. Our implementation is available at https://github.com/Dshijie/PIMVC.

Graph-based multi-view clustering has attracted extensive attention because of the powerful clustering-structure representation ability and noise robustness. Considering the reality of a large amount of incomplete data, in this paper, we propose a simple but effective method for incomplete multi-view clustering based on consensus graph learning, termed as HCLS_CGL. Unlike existing methods that utilize graph constructed from raw data to aid in the learning of consistent representation, our method directly learns a consensus graph across views for clustering. Specifically, we design a novel confidence graph and embed it to form a confidence structure driven consensus graph learning model. Our confidence graph is based on an intuitive similar-nearest-neighbor hypothesis, which does not require any additional information and can help the model to obtain a high-quality consensus graph for better clustering. Numerous experiments are performed to confirm the effectiveness of our method.

In recent years, many incomplete multi-view clustering methods have been proposed to address the challenging unsupervised clustering issue on the multi-view data with missing views. However, most of the existing works are inapplicable to large-scale clustering task and their clustering results are unstable since these methods have high computational complexities and their results are produced by kmeans rather than their designed learning models. In this paper, we propose a new one-step incomplete multi-view clustering model, called Localized and Balanced Incomplete Multi-view Clustering (LBIMVC), to address these issues. Specifically, LBIMVC develops a new graph regularized incomplete multi-matrix-factorization model to obtain the unique clustering result by learning a consensus probability representation, where each element of the consensus representation can directly reflect the probability of the corresponding sample to the class. In addition, the proposed graph regularized model integrates geometric preserving and consensus representation learning into one term without introducing any extra constraint terms and parameters to explore the structure of data. Moreover, to avoid that samples are over divided into a few clusters, a balanced constraint is introduced to the model. Experimental results on four databases demonstrate that our method not only obtains competitive clustering performance, but also performs faster than some state-of-the-art methods.

In this article, we aim to design a distributed approximate algorithm for seeking Nash equilibria (NE) of an aggregative game. Due to the local set constraints of each player, projection-based algorithms have been widely employed for solving such problems actually. Since it may be quite hard to get the exact projection in practice, we utilize inscribed polyhedrons to approximate local set constraints, which yields a related approximate game model. We first prove that the NE of the approximate game is the ϵ -NE of the original game and then propose a distributed algorithm to seek the ϵ -NE, where the projection is then of a standard form in quadratic optimization with linear constraints. With the help of the existing developed methods for solving quadratic optimization, we show the convergence of the proposed algorithm and also discuss the computational cost issue related to the approximation. Furthermore, based on the exponential convergence of the algorithm, we estimate the approximation accuracy related to ϵ . In addition, we investigate the computational cost saved by approximation in numerical simulation.