Signed-Directed graphs, known for the ability to express complex relationships, has been the most valuable among several graph types. However, real-world data often contains severe noise and complexity, making it challenging to analyze. Graph contrastive learning, a powerful technique for learning discriminative representations, has become an essential tool for various graph mining tasks. Therefore, we explore graph contrastive learning with signed-directed graphs. To provide multi-scale discriminative representations, we apply framelet transforms to create more intricate filtering representations and propose a Magnetic Framelet-Based Graph Contrastive Learning for Signed-Directed graphs (Framelet-Gcl). This approach offers a robust model for complex graph data analysis by employing structure and Laplacian perturbations to construct node-level contrastive loss. The overall architecture performs framelet-based convolution in both real and complex domains for each view, enhancing the basis for signal processing. Experimental results demonstrate that the proposed approach outperforms existing state-of-the-art methods across various evaluation metrics on four real-world datasets.

Graph Neural Networks (GNNs) have shown great performance in learning representations for graph-structured data. However, recent studies have found that the interference between topology and attribute can lead to distorted node representations. Most GNNs are designed based on homophily assumptions, thus they cannot be applied to graphs with heterophily. This research critically analyzes the propagation principles of various GNNs and the corresponding challenges from an optimization perspective. A novel GNN called Graph Neural Networks with Soft Association between Topology and Attribute (GNN-SATA) is proposed. Different embeddings are utilized to gain insights into attributes and structures while establishing their interconnections through soft association. Further as integral components of the soft association, a Graph Pruning Module (GPM) and Graph Augmentation Module (GAM) are developed. These modules dynamically remove or add edges to the adjacency relationships to make the model better fit with graphs with homophily or heterophily. Experimental results on homophilic and heterophilic graph datasets convincingly demonstrate that the proposed GNN-SATA effectively captures more accurate adjacency relationships and outperforms state-of-the-art approaches. Especially on the heterophilic graph dataset Squirrel, GNN-SATA achieves a 2.81% improvement in accuracy, utilizing merely 27.19% of the original number of adjacency relationships. Our code is released at https://github.com/wwwfadecom/GNN-SATA.

Structured clustering networks, which alleviate the oversmoothing issue by delivering hidden features from autoencoder (AE) to graph convolutional networks (GCNs), involve two shortcomings for the clustering task. For one thing, they used vanilla structure to learn clustering representations without considering feature and structure corruption; for another thing, they exhibit network degradation and vanishing gradient issues after stacking multilayer GCNs. In this article, we propose a clustering method called dual-masked deep structural clustering network (DMDSC) with adaptive bidirectional information delivery (ABID). Specifically, DMDSC enables generative self-supervised learning to mine deeper interstructure and interfeature correlations by simultaneously reconstructing corrupted structures and features. Furthermore, DMDSC develops an ABID module to establish an information transfer channel between each pairwise layer of AE and GCNs to alleviate the oversmoothing and vanishing gradient problems. Numerous experiments on six benchmark datasets have shown that the proposed DMDSC outperforms the most advanced deep clustering algorithms.

Probabilistic linear discriminant analysis (PLDA) is a very effective feature extraction approach and has obtained extensive and successful applications in supervised learning tasks. It employs the squared$L_{2}$-norm to measure the model errors, which assumes a Gaussian noise distribution implicitly. However, the noise in real-life applications may not follow a Gaussian distribution. Particularly, the squared$L_{2}$-norm could extremely exaggerate data outliers. To address this issue, this article proposes a robust PLDA model under the assumption of a Laplacian noise distribution, called L1-PLDA. The learning process employs the approach by expressing the Laplacian density function as a superposition of an infinite number of Gaussian distributions via introducing a new latent variable and then adopts the variational expectation–maximization (EM) algorithm to learn parameters. The most significant advantage of the new model is that the introduced latent variable can be used to detect data outliers. The experiments on several public databases show the superiority of the proposed L1-PLDA model in terms of classification and outlier detection.

The objective of self-expression based spectral clustering is to learn an affinity matrix which accurately reflects the similarity among data, and the Laplacian constraint is usually exploited to make the affinity matrix preserve the global structure of raw data. However, there exist two drawbacks: firstly, these methods are mostly designed for vectorial data in Euclidean spaces, which are not suitable for multidimensional data with nonlinear manifold structure, e.g., videos and image-sets. Secondly, the clustering performance heavily relies on the quality of a pre-learned Laplacian matrix in which the global structure may be mis-interpreted without considering manifold structures. In this paper, we firstly provide a unified framework about self-expression learning on Grassmann manifolds, which implements the clustering tasks for multidimensional data under subspace views. Then, to assign optimal neighbors to each data depending on the local distance, we adaptively learn the neighborhood relationship from the obtained self-expression coefficient matrix, referred to Learning Adaptive Neighborhood Graph on Grassmann manifolds (GMAN). In the optimization process, the neighborhood relationship can be adaptively learned and updated with the coefficient matrix. The experimental results on five public datasets show that the proposed method is obviously better than many related clustering methods based on Grassmann manifolds, proving the effectiveness of GMAN in multidimensional data clustering.

Multiview clustering (MVC) has recently received great interest due to its pleasing efficacy in combining the abundant and complementary information to improve clustering performance, which overcomes the drawbacks of view limitation existed in the standard single-view clustering. However, the existing MVC methods are mostly designed for vectorial data from linear spaces and, thus, are not suitable for multiple dimensional data with intrinsic nonlinear manifold structures, e.g., videos or image sets. Some works have introduced manifolds' representation methods of data into MVC and obtained considerable improvements, but how to fuse multiple manifolds efficiently for clustering is still a challenging problem. Particularly, for heterogeneous manifolds, it is an entirely new problem. In this article, we propose to represent the complicated multiviews' data as heterogeneous manifolds and a fusion framework of heterogeneous manifolds for clustering. Different from the empirical weighting methods, an adaptive fusion strategy is designed to weight the importance of different manifolds in a data-driven manner. In addition, the low-rank representation is generalized onto the fused heterogeneous manifolds to explore the low-dimensional subspace structures embedded in data for clustering. We assessed the proposed method on several public data sets, including human action video, facial image, and traffic scenario video. The experimental results show that our method obviously outperforms a number of state-of-the-art clustering methods.

Restricted Boltzmann machine (RBM) is a famous model for feature extraction and can be used as an initializer for neural networks. When applying the classic RBM to multidimensional data such as 2D/3D tensors, one needs to vectorize such as high-order data. Vectorizing will result in dimensional disaster and valuable spatial information loss. As RBM is a model with fully connected layers, it requires a large amount of memory. Therefore, it is difficult to use RBM with high-order data on low-end devices. In this article, to utilize classic RBM on tensorial data directly, we propose a new tensorial RBM model parameterized by the tensor train format (TTRBM). In this model, both visible and hidden variables are in tensorial form, which are connected by a parameter matrix in tensor train format. The biggest advantage of the proposed model is that TTRBM can obtain comparable performance compared with the classic RBM with much fewer model parameters and faster training process. To demonstrate the advantages of TTRBM, we conduct three real-world applications, face reconstruction, handwritten digit recognition, and image super-resolution in the experiments.

Linear discriminant analysis (LDA) has been a widely used supervised feature extraction and dimension reduction method in pattern recognition and data analysis. However, facing high-order tensor data, the traditional LDA-based methods take two strategies. One is vectorizing original data as the first step. The process of vectorization will destroy the structure of high-order data and result in high dimensionality issue. Another is tensor LDA-based algorithms that extract features from each mode of high-order data and the obtained representations are also high-order tensor. This paper proposes a new probabilistic LDA (PLDA) model for tensorial data, namely, tensor PLDA. In this model, each tensorial data are decomposed into three parts: the shared subspace component, the individual subspace component, and the noise part. Furthermore, the first two parts are modeled by a linear combination of latent tensor bases, and the noise component is assumed to follow a multivariate Gaussian distribution. Model learning is conducted through a Bayesian inference process. To further reduce the total number of model parameters, the tensor bases are assumed to have tensor CandeComp/PARAFAC (CP) decomposition. Two types of experiments, data reconstruction and classification, are conducted to evaluate the performance of the proposed model with the convincing result, which is superior or comparable against the existing LDA-based methods.

Restricted Boltzmann Machine (RBM) is a particular type of random neural network models modeling vector data based on the assumption of Bernoulli distribution. For multidimensional and non-binary data, it is necessary to vectorize and discretize the information in order to apply the conventional RBM. It is well-known that vectorization would destroy internal structure of data, and the binary units will limit the applying performance due to fickle real data. To address these issues, this paper proposes a Matrix variate Gaussian Restricted Boltzmann Machine (MVGRBM) model for matrix data whose entries follow Gaussian distributions. Compared with some other RBM algorithms, MVGRBM can model real value data better and it has good performance in image classification. To prove that adding Gaussian parameters could model input data well, we compared the reconstruction performance of the Gaussian parameters updating and fixed.

The probabilistic principal component analysis (PPCA) is built upon a global linear mapping, with which it is insufficient to model complex data variation. This paper proposes a mixture of bilateral-projection probabilistic principal component analysis model (mixB2DPPCA) on 2D data. With multi-components in the mixture, this model can be seen as a 'soft' cluster algorithm and has capability of modeling data with complex structures. A Bayesian inference scheme has been proposed based on the variational EM (Expectation-Maximization) approach for learning model parameters. Experiments on some publicly available databases show that the performance of mixB2DPPCA has been largely improved, resulting in more accurate reconstruction errors and recognition rates than the existing PCA-based algorithms.

In image restoration problems, it is reasonable to add nonnegative constraints because of the physical meaning of images. In general, this problem can be expressed as a quadratic programming problem with nonnegative constraints, which results in a linear complementary problem from the KKT optimization conditions. By reformulating the linear complementary problem as implicit fixed-point equations, a class of modulus-based matrix splitting iteration methods is established. In this paper, for a better computational implementation, we present an inexact iteration process for these modulus-based methods. Convergence properties for this inexact process are analyzed, and some specific implementations for the inner iterations are presented. Numerical experiments for nonnegatively constrained image restorations are presented, and the results show that our methods are comparable and more efficient than the existing projection type methods.

This paper introduces an L1-norm-based probabilistic principal component analysis model on 2D data (L1-2DPPCA) based on the assumption of the Laplacian noise model. The Laplacian or L1 density function can be expressed as a superposition of an infinite number of Gaussian distributions. Under this expression, a Bayesian inference can be established based on the variational expectation maximization approach. All the key parameters in the probabilistic model can be learned by the proposed variational algorithm. It has experimentally been demonstrated that the newly introduced hidden variables in the superposition can serve as an effective indicator for data outliers. Experiments on some publicly available databases show that the performance of L1-2DPPCA has largely been improved after identifying and removing sample outliers, resulting in more accurate image reconstruction than the existing PCA-based methods. The performance of feature extraction of the proposed method generally outperforms other existing algorithms in terms of reconstruction errors and classification accuracy.