Multi-view semi-supervised learning leverages consistent and complementary patterns under limited supervision. However, a critical challenge arises from the inherent optimization conflict between shared semantic alignment and unique complementarity, where neglecting this entanglement induces representation ambiguity and accumulates noise during the training process. To address these issues, this paper proposes a unified framework with information-aware feature decomposition and reliable multi-space prototypes (IDMP). The information-aware decomposition explicitly decomposes the embedding feature into view-shared semantic features and view-unique complementarity information, forming two functionally orthogonal components. Moreover, we introduce multi-space prototypes to enhance feature discriminability for task-oriented learning, complemented by a multi-consistency criterion module that refines pseudo-labels to ensure stable convergence. Experiments demonstrate that the proposed method outperforms baselines in accuracy and F1-score on multiple benchmark datasets, validating its superiority.

Existing multi-view feature fusion methods exhibit limitations in modeling high-order correlations, particularly the underutilization of matrix-structured representations, ineffective redundancy elimination, and insufficient deep model integration. To address these critical limitations, we propose a high-order relation-driven framework for interpretable multi-view feature fusion. To obtain accurate per-view representations, we first distill latent embeddings in a pair of complementary graph-structured spaces. These embeddings are then stacked into a third-order tensor, thereby enabling a unified characterization of global cross-view high-order interactions. Redundancy across tensor slices is subsequently reduced by enforcing a low-rank structure under the tensor-SVD nuclear norm, preserving discriminative information while eliminating duplicate components. Finally, feature separability is enhanced via a discriminative loss that explicitly enforces view-specific uniqueness constraints. Extensive experiments on eleven public datasets establish state-of-the-art multi-view clustering and classification performance, surpassing thirteen representative baselines. The resultant interpretable network, entirely free of parameter tuning, further enhances model transparency. The code is available at https://github.com/Elena365-star/HoRM.

Multi-view fusion has emerged as an effective paradigm for learning unified embeddings from heterogeneous sources. Graph-based methods have gained recognition for their capability to capture and align cross-view structural relationships among instances. Currently, in the unsupervised setting, graph neural network-based multi-view learning faces three key challenges: constructing reliable graph topologies, aligning inter-view relationships, and exploiting structural information. In this paper, we propose the Global Graph Contrastive learning for Multi-view fusion (G2CM) algorithm to address these issues. To construct reliable graph topologies, we integrate a global topology with view-specific weighted edges, where the global topology encodes relational context and the view-specific weighting preserves local semantics within each view. These weighted adjacency matrices are then used to build individual graph convolutional networks for each view, allowing more precise modeling and integration of global and local structural information during representation learning. To enhance cross-view alignment, we introduce a contrastive learning framework with three types of positive pairs and two types of negative pairs to capture multi-level semantic relationships across views. By jointly modeling intra-view and inter-view alignments, the framework enhances the discriminative capability and semantic consistency of the learned embeddings. To better exploit structural information, we incorporate distance-aware scaling into the loss function, weighting sample pairs by their feature-space proximity. This mechanism enhances the influence of semantically relevant neighbors during optimization, improving the preservation of local proximity and utilization of structural information. Experiments on six benchmark multi-view datasets demonstrate that G2CM achieves state-of-the-art performance across diverse data types.

As a special category of GNNs, implicit GNNs can capture the long-range dependencies of nodes and reduce memory consumption by transforming forward propagation into a fixed-point iterative process. However, these implicit graph models confront two pressing issues. First, they lack a systematic discussion on the effect of feature perturbations, which is critical given recent studies showing explicit GNNs' vulnerability to adversarial attacks in socially contextualized graphs. Moreover, existing models based on unconstrained homogeneous differential equations may lead to degradation of related representations. Inspired by error propagation in dynamic systems, we discover a robust intrinsic layer hidden in such systems and construct a robust graph neural network with implicit layers. Theoretical analysis shows that certain constraints ensure the model's robustness against feature perturbations. Furthermore, we construct a non-homogeneous ordinary differential equation with parameter constraints, leading to more stable and reliable solutions. Extensive experiments, including semi-supervised node classification and feature perturbation defense on multiple datasets, demonstrate the superiority of the proposed method.

In recent years, Graph Neural Networks (GNNs) have emerged as a powerful tool for learning on graph-structured data, achieving significant success across various fields. Most existing GNNs follow a message passing paradigm that iteratively aggregates features from adjacent nodes to update node representations. However, in low-degree graphs, the limited number of adjacent nodes makes it challenging to fully explore the complex relationships between nodes during the information aggregation process, leading to suboptimal performance in graph learning tasks. Besides, augmentation methods primarily focus on enhancing the richness of node neighborhoods,whichs overlooks the need for GNNs to capture complex relationships during the aggregation process. To this end, this paper introduces a framework for architecture-agnostic data augmentation, termed Multi-level Augmentation (MuLA). The framework consists of three core modules: (1) The graph representation augmentation module enriches the information representation of low-degree nodes by combining their features with those of their unlabeled neighbors; (2) The graph knowledge integration module further enhances node representations by learning the augmented neighborhood information and integrating it with the original features; (3) The Task-Adaptive module dynamically adjusts the framework to accommodate various machine learning tasks. Experimental results demonstrate that the proposed MuLA significantly improves the performance of existing GNNs on semi-supervised node classification and link prediction tasks across multiple datasets, validating its effectiveness and broad applicability.

Multi-view graph learning has garnered increasing attention for its ability to leverage complementary structural and semantic information across views. However, most existing methods primarily focus on intra-view information propagation, while overlooking complex inter-view interactions. To better extract higher-order relationships between views, we propose a unified framework that dynamically extracts inter-view information by aggregating positional encoding information across views. First, view-specific structural signatures are derived and fused via a similarity-guided projection strategy, enabling the model to capture both common and distinctive topological patterns across views. Second, a dynamic degree encoding scheme is introduced to incorporate node centrality into the message-passing mechanism, thereby enhancing structural awareness. Third, a hierarchical fusion module, incorporating attention and gating mechanisms, is proposed to adaptively integrate multi-view node representations under global contextual guidance and feature-wise modulation. Furthermore, to foster both consistency and discrimination among views, a contrastive learning objective is incorporated to align cross-view representations within a latent space. Extensive experiments on multiple benchmark datasets demonstrate that our method consistently outperforms state-of-the-art multi-view graph learning approaches, achieving superior accuracy, robustness, and generalization.

Recently, graph-aware Large Language Models (LLMs) have shown promising capabilities in jointly modeling graph-structured data and textual information. Existing approaches typically employ a graph encoder and a frozen LLM to obtain node representations from graph and textual views, followed by node-level alignment to bridge the two modalities. However, such alignment mechanisms primarily focus on node information while overlooking edge-level structures, leading to suboptimal information propagation across views. In this work, we conduct a comprehensive theoretical analysis to uncover why node-level alignment is insufficient for aligning textual and graph representations. Specifically, we prove theoretically for the first time that neglecting edge information leads to suboptimal solutions and negatively curved edges induce bottlenecked information flow, giving rise to the over-squashing phenomenon between graph and textual views. To address the two challenges, we innovatively proposed a CureLLM framework of Curvature-enhanced Graph Representations for Large Language Model whose goal is to inject the signals of edge information into the existing LLMs. Specifically, CureLLM first introduces the training-free textual prompt mechanism to make the LLM model generate the output directly based on the edge-aware prompt without learnable parameter costs. Furthermore, a novel curvature-aware graph representation learning is designed to capture the edge structure information to enhance the downstream tasks, where the message passing between text and graph representations only depends on edges with positive curvature. Finally, we conduct evaluations with 20 different compared methods on 11 real world datasets from various domains and the experiment results demonstrate the superiority of our proposed CureLLM framework.

User preference modeling based on hypergraphs has shown significant potential in recommender systems. However, existing methods model complex higher-order relations rely on existing hypergraph structures, such well-constructed hypergraphs are not readily accessible in every situation. Furthermore, since existing methods perform message-passing based on the same hypergraph convolution function, they often overlook diverse relation patterns, thus lacking precision. In this work, we propose an Enhanced Recommendation Framework with Hypergraph Mixture of Experts (HMoRec). Specifically, we first employ a sparse optimal transport clustering mechanism to generate high-quality hypergraph without requiring external knowledge. Then, we model diverse higher-order interactions and enhance representation learning based on the hypergraph mixture of experts and cross-view representation fusion. Extensive experiments on four real-world multi-domain datasets have shown that our HMoRec achieves significant performance gains.

Multi-view learning has attracted considerable attention for its ability to enhance model performance by fusing complementary information from multiple data modalities. Among them, multi-view approaches based on graph convolutional networks have achieved remarkable progress in this field. However, existing methods are often constrained by fixed propagation layers and static fusion mechanisms, which limit their ability to adapt to the dynamic nature of graph data and capture long-range dependencies effectively. These limitations impede the model’s ability to dynamically adapt to the evolving relationships and interactions between views, thereby restricting its capacity to fully exploit the complementary information embedded within multi-view datasets. To address these challenges, we propose a framework that integrates ordinary differential equations with graph convolutional networks. First, this framework learns a unified latent representation through adaptive feature transformation and a cross-view consistency-preserving module, which jointly optimizes view alignment and feature mapping via residual connections and inter-view regularization. Then the proposed framework leverages a dynamic ODE-based propagation mechanism to model feature evolution, allowing for continuous dynamic representation of node attributes while adaptively adjusting propagation depth to mitigate over-smoothing and capture both short- and long-range dependencies. Extensive experiments on benchmark multi-view datasets demonstrate that our approach consistently outperforms state-of-the-art methods in terms of classification accuracy and robustness, particularly under noisy or highly diverse view conditions.

Multi-view learning based on graph convolutional networks can capitalize on supervisory information from heterogeneous data views to improve performance, thereby attracting increasing attention in various practical fields. Although existing research and applications have made significant progress, they mostly consider feature and topological structure learning separately, resulting in underutilization of learned representations or topological structures and susceptibility to noise. To address these challenges, we first propose a bi-level optimization framework for an interpretable graph neural network in an end-to-end manner, then introduce adversarial training into multi-view learning, modeling the multi-view adversarial graph convolutional problem as a minimax optimization problem. Specifically, the lower-level optimization maximizes graph regularization within a robust control range to refine the graph Laplacian, while the upper-level minimizes the loss caused by the lower-level perturbations to enhance representation robustness. This jointly improves model adaptability to noisy data. The proposed approach aims to synchronize representation learning and topological adjustment to ensure effective information transfer between features and topological structures. It represents an efficient solution for multi-view graph convolutional networks within the dynamic graph structure learning paradigm. Extensive experiments validate our method’s effectiveness across diverse real-world multi-view datasets.

Multi-source data is now widely available, yet sensor and device failures still cause frequent data loss, leading to the challenging problem of incomplete multi-view clustering. Although some graph-based methods perform well, many only construct view-specific graphs and thus ignore graph misalignment induced by differing missing patterns across views. Others attempt to learn a consensus representation via graphs but rely solely on fully observed samples, resulting in insufficient data utilization problem. To address the above issues, we propose a deep unfolding framework for consensus representation and graph learning in incomplete multi-view clustering. Specifically, the proposed method leverages sample-view observation indicator matrices to build the consensus graph, mitigating the spread of incorrect neighbor relationships caused by missing data. Additionally, the framework consists of multiple deep unfolding layers, each containing several independently differentiable variables including view-specific projection matrices that preserve the unique information of each view via closed-form solutions, a consensus graph that constructs the shared similarity graph through similarity aggregation, and a consensus representation that updates the latent representation via proximal gradient descent. Extensive experiments on public datasets validate the effectiveness of our method in handling incomplete multi-view clustering task.

With the rapid development of multimedia technologies and the explosive growth of multi-source heterogeneous data, multi-view learning has attracted increasing attention. Although existing methods have effectively modeled intra-view and inter-view relationships, inevitable noise in real-world scenarios still leads to imbalanced view quality, which in turn introduces conflicts during view integration and causes decision bias in the fused representations. To address these challenges, we propose a reliability-aware dual graph convolutional network to enhance the reliability of view fusion. Specifically, we introduce a reliability-guided Bayesian inference mechanism to characterize view quality and learn shared representations across views under the guidance of view evidence. Moreover, to comprehensively capture both intra-view and inter-view data dependencies, we propose a dual graph convolutional network that collaboratively models intra-view complementarity and inter-view consistency, thereby improving robustness and overall learning stability. Experiments on multiple benchmark datasets demonstrate the effectiveness and robustness of the proposed method for multi-view semi-supervised classification.

Multi-view learning constructs robust joint representations by effectively integrating consistent and complementary information from heterogeneous sources. Graph neural networks have achieved substantial progress in multi-view learning. However, most existing approaches primarily rely on localized message-passing mechanisms to capture local topological structures, while insufficiently modeling heterogeneous node relationships and their varying influence. In this paper, we propose multi-relational graph Transformer for multi-view learning. This architecture integrates multi-view information to model complex node associations by constructing multi-relational graphs, where samples are explicitly regarded as nodes to encode heterogeneous relationships. A relational graph Transformer is designed to learn the impact of heterogeneous relationships on feature propagation, yielding more discriminative node representations. Additionally, this method introduces learnable multi-relation fusion to integrate relation propagated messages with original node representations and combine relation-specific features. Extensive experiments on multiple benchmark datasets demonstrate the effectiveness of the proposed method for multi-view semi supervised node classification, achieving an average accuracy improvement of 10.76% over the baseline methods. Notably, on the class imbalanced Prokaryotic dataset, our approach attains a 5.52% improvement over the best-performing baseline, further showing its effectiveness and applicability.

Data missing is a common issue in real-world applications, posing significant challenges for incomplete data processing. Traditional incomplete multi-view clustering methods rely on manually-designed optimization problems based on prior interpretable knowledge, considering the full utilization of available data. However, their limited feature extraction capability may become a bottleneck. In contrast, deep optimization methods leverage learning-based nonlinear transformations for clustering. They primarily achieve data imputation through the generalization ability of deep models, but their model interpretability may be limited by the black-box nature. Moreover, most existing methods only explore the structure of each view independently, where these structures are fixed and cannot form a complete unified structure. To address these issues, we propose a Structural Optimization-inspired Interpretable Network (SOI-Net) for incomplete multi-view clustering. Specifically, we project the features of all views into a unified representation space with the un-missing information of the views as constraints. By optimizing consistent structural information, we preserve the structures of missing modalities in the unified representation space, thereby mitigating the impact of missing data. Meanwhile, we derive network components based on the optimization problem to guide the learning of structure and representation. The practical significance of these network components provides model design-level interpretability. Extensive experiments on six datasets validate the effectiveness of SOI-Net in handling incomplete multi-view clustering task.

Temporal knowledge graph forecasting (TKGF) has become a crucial tool for predicting future facts based on temporal knowledge graphs. Traditional methods face inherent limitations: rule-based reasoning approaches are restricted to exploring events that are related to historical rules and fail to fully utilize global graph information. In contrast, although embedding-based methods have the ability to capture global graph information through vector representations, they struggle to effectively incorporate historical rules. To address these issues, this article proposes a novel framework that combines historical rule-based reasoning with embedding-based prediction enhanced by curriculum learning (CL) to predict unknown events. The framework consists of two complementary modes: the rule mode, which extracts temporal rules through time-filtered weighted sampling and applies them for prediction, and the embedding mode, which progressively represents entities, relations, and time as vectors from simple to complex through CL and generates predictions based on these embedding vectors. The framework effectively integrates rules with the entire graph information by combining the predictions from two models, resulting in a more comprehensive prediction. Extensive experiments on multiple public datasets demonstrate that the proposed method outperforms state-of-the-art approaches in terms of performance, validating the effectiveness of its hybrid approach, and highlighting the significant potential of CL in TKGF.

Representation learning techniques effectively unveil latent patterns within raw data. However, the learning process is often marred by uncertainties, such as variations in data quality and heterogeneous scenarios, which greatly affect the reliability of representation learning. In this article, we introduce a reliable representation learning framework to establish a connection between data attributes and modeling strategies, namely the interpretable attribute-oriented representation learning framework. First, by focusing on the inherent knowledge embedded in the data, we decouple it into four principal attributes: fidelity, topology, invariance, and discriminability. To explicitly address these attributes, we incorporate them into an optimization-derived framework using corresponding general loss terms. Furthermore, by treating the iterative solution process as a bridge, each derived network module possesses traceable interpretability, thus laying a reliable foundation. Ultimately, we extend the proposed framework to multisource heterogeneous scenarios, enabling it to adapt to complex environments while maintaining reliability. In essence, our work aims to seamlessly integrate deep representations with prior knowledge during the learning process, thereby creating a solid basis for dependable modeling. Networks derived from the proposed framework achieve promising results, particularly in complex multisource heterogeneous environments, demonstrating both their effectiveness and reliability. The code is available at https://github.com/ZihanFang11/2025_AORLNet_TNNLS.

Traditional multi-view learning models are primarily designed for static datasets with fixed views. However, in dynamic incremental view environments, this approach inevitably leads to view forgetting, where the introduction of new views weakens previously acquired knowledge. In contrast, the human brain exhibits remarkable memory retention and knowledge transfer capabilities when receiving objects described from different perspectives, with past experience further supplementing new insights. Inspired by underlying neural processing mechanisms, we propose a novel view incremental learning framework named Hebbian View Orthogonal Projection (HVOP). HVOP constructs a knowledge transfer space, where gradient updates are projected onto the orthogonal complement of historical representations, thereby mitigating interference between old and new views. By further incorporating recursive lateral connections and Hebbian learning rules, the proposed model imparts brain-like dynamic adaptability to the learning process, enhancing knowledge transfer and integration, thereby enabling stable knowledge transfer under evolving views. We validate HVOP on node classification tasks, demonstrating its superior performance in both knowledge retention and transfer compared to traditional methods. The results highlight the efficacy of biologically inspired mechanisms in mitigating the view forgetting phenomenon.

Label scarcity remains a core challenge in graph-based learning, especially in text-attributed graphs (TAGs), where entangled node semantics further increase annotation difficulty. Recently, Large Language Models (LLMs) have emerged as a promising alternative oracle, but their predictions are inherently noisy and tend to be overconfident. In this paper, we propose a confidence-calibrated node selection framework that explicitly models and harnesses the inherent characteristics of noisy LLM annotations. Specifically, we estimate annotation difficulty via soft assignment to construct a diverse candidate pool while controlling annotation cost. Through empirical analysis of noisy LLM annotations, we approximate the underlying confusion pattern using a prototype-based confusion matrix, which captures inter-class ambiguity and provides an interpretable global view of confusion patterns. Building on this insight, we introduce a graph-LLM confidence calibration module, which jointly models global label reliability and performs graph-aware propagation to adjust confidence scores based on neighborhood label distributions. Extensive experiments on real-world datasets demonstrate that the proposed method significantly outperforms baselines, providing a principled approach to harnessing noisy LLM annotations.

Deep multi-view representation learning has been recognized as a vital research direction owing to its strong ability to fuse and interpret complex data patterns. Nevertheless, the intrinsic opacity inherent in deep neural network architectures significantly hampers interpretability, particularly regarding mechanisms for the integration of heterogeneous information across multiple views. To bridge this gap, we propose an interpretable deep unfolding method tailored specifically for multi-view representation learning. Initially, we establish the representation learning task as a generalized optimization problem that integrates essential multi-view priors, including semantic consistency, spatial topology, noise diversity, and informational complementarity. Building on this conceptual groundwork, we leverage the Alternating Direction Method of Multipliers to derive iterative solutions, subsequently embedding these solutions into a deep representation learning architecture. This framework rigorously adheres to an iterative unfolding paradigm, explicitly incorporating four distinct categories of multi-view prior knowledge, thereby augmenting the interpretability embedded in the network’s design while concurrently facilitating efficient iterative optimization. Empirical analyses using four exemplar models instantiated from the proposed methods reveal superior performance across challenging multi-view tasks, substantiating the framework’s fusion efficacy and interpretability.

Graph Neural Networks (GNNs) have become a powerful tool for learning representations from graph-structured data, leveraging the relationships between nodes and their features. Despite their success, they often lack interpretability due to the black-box nature of neural networks, and further development may be limited. Moreover, previous GNN-based multi-view methods typically rely on simple feature fusion techniques such as weighted averaging or concatenation, which fail to capture the complex dependencies between views. In this paper, we propose a novel framework, namely Interpretable Multi-View Feature Representation via physical partial differential equation (IMvFR), to address these limitations in the context of multi-view semi-supervised learning. By integrating GNNs with partial differential equations (PDEs), we model the evolution of multi-view feature representations as a dynamic process. This provides a natural and interpretable framework for understanding how information flows between different views, overcoming the black-box nature of traditional GNNs. Additionally, we formulate multi-view feature representations as an initial-value problem within the framework of PDEs, providing a clear and interpretable mechanism for label propagation and feature fusion, thus facilitating the acquisition of global and local information between views. Comprehensive experimental results on eight datasets demonstrate that the proposed method achieves superior performance compared with state-of-the-art methods.