Modern multi-view clustering (MVC) is dominated by two paradigms: multi-view fusion and pseudo-label-guided learning. Pseudo-labeling methods can suffer from confirmation bias; their reliance on a fixed-granularity supervision from an initial clustering can cause learned embeddings to drift from the data's true structure and lose discriminative power. Conversely, fusion methods excel at integrating information but often struggle to robustly differentiate between high-quality and noisy views, which can obscure final cluster boundaries and degrade performance. To address these complementary challenges, we propose GAPS (Granularity-Aware Pseudo Supervision), a novel MVC framework. GAPS introduces a granularity-aware supervision mechanism that generates a full hierarchy of pseudo-labels, enabling the selection of a supervision level that best aligns with the data's intrinsic multi-scale structure. Furthermore, to ensure a high-quality supervisory signal, it incorporates a reliability-aware view selection strategy using a novel Separation-Compactness Index (SCI) to identify and leverage the most informative view for pseudo-label generation. This dual approach ensures the supervisory signal is both structurally adaptive and derived from the most reliable source, leading to highly effective final representations. Extensive experiments on synthetic and real-world datasets demonstrate the effectiveness and superiority of GAPS over other competitors.

Clustering on high-dimensional and low sample size (HDLSS) data remains a critical, persistent challenge where extreme sparsity and noise confound cluster analysis. This creates a dilemma: spectral methods fail as distance metrics degrade, while deep clustering tends to over-fit scarce data. To break this dilemma, a Self-Enhanced Density Clustering (SEDC) framework that integrates the cluster structure discovery and embedding representation learning into an iterative enhancement process is proposed in this paper. Specifically, SEDC uses adaptive density-derived centroids to parameterize probabilistic soft labels, which in turn supervise a lightweight multilayer perceptron (MLP) to learn the low-dimensional embedding from data. The resulting embedding provides a refined metric space for further generating superior labels in the subsequent interaction process. This feedback forms a mutual reinforcement that progressively enhances the discrimination of embedding while rigorously mitigating over-fitting. Extensive experiments on 43 challenging HDLSS datasets demonstrate state-of-the-art performance, substantially outperforming popular clustering methods. This work delivers a principled and promising solution for robust data clustering in HDLSS situations.

Most existing outlier detection methods rely on a single and fine-grained data representation, making them vulnerable to noise and inefficient in capturing local anomalies. Granular-ball computing (GBC), as an emerging multi-granularity representation and computation framework, provides an effective means to address these issues. Meanwhile, shadow set theory offers a flexible mechanism using three-way decision to handle uncertainty and boundary fuzziness in data. The integration of GBC with shadow set theory combines the strengths of both frameworks, offering promising potential for outlier detection tasks. In this study, we propose a novel outlier detection based on shadowed granular-balls. Firstly, we propose an unsupervised granular-ball generation method with the principle of justifiable granularity. Then, we further present an outlier detection method, named three-way outlier detection based on shadowed granular-ball (3W-SGBD). 3W-SGBD introduces an unsupervised granular-ball generation strategy guided by local density clustering, and adaptively splits granular-balls through a dual-entropy-driven mechanism to better capture local anomalies. In addition, by partitioning each granular-ball into positive, negative, and boundary regions via shadow mapping, 3W-SGBD refines boundary areas to enhance detection accuracy. Finally, extensive comparative experiments are conducted with several state-of-the-art baseline methods on 16 public benchmark datasets. The results show that the effectiveness, efficiency, and robustness of the method proposed in this paper. The code is publicly available athttps://github.com/2257352568/3W-SGBD.

As partial samples are often absent in certain views, incomplete multi-view clustering has become a challenging task. To tackle data with missing views, current methods either utilize the data similarity relations to recover missing samples or primarily consider the available information of existing samples, typically facing some inherent limitations. Firstly, traditional solutions cannot fully explore the potential information contained in missing samples due to their omission strategy, leading to sub-optimal graphs. Moreover, most methods mainly focus on data recovery from the view level, ignoring the differences among available/missing samples in various views. To this end, we propose a collaborative Similarity Fusion and Consistency Recovery (SFCR) method, which resolves the incomplete multi-view clustering problem by learning a unified similarity graph and recovering missing samples with consistent structures. Specifically, to learn a reliable graph compatible across views, a novel view-to-sample fusion model is designed to adaptively coalesce the view-wise similarities among available samples, not only preserving the complementarity and consistency among views but also properly balancing different samples. Furthermore, the missing samples are effectively recovered under the guidance of the fused similarity graph, so as to maintain the consistent structure of recovered data across views. In this way, the similarity learning and the missing data recovery benefit from each other in a collaborative reinforcement manner. Meanwhile, SFCR can directly obtain the final clustering labels without additional post-processing. Extensive experiments demonstrate the effectiveness and superiority of SFCR.

As a foundational clustering paradigm, Density Peak Clustering (DPC) partitions samples into clusters based on their density peaks, garnering widespread attention. However, traditional DPC methods usually focus on high-density regions, neglecting representative peaks in relatively low-density areas, particularly in datasets with varying densities and multiple peaks. Moreover, existing DPC variants struggle to identify clusters correctly in high-dimensional spaces due to the indistinct distance differences among samples and sparse data distributions. Additionally, existing methods typically adopt a one-step label assignment strategy, making them prone to cascading errors when initial misassignments occur. To address these challenges, we propose an Enhanced Density Peak Clustering (EDPC) method, which creatively incorporates multilayer perceptron (MLP)-based dimensionality reduction and a hierarchical label assignment strategy to significantly improve clustering performance in high-dimensional scenarios. Specifically, we introduce an effective selection condition that combines average densities and density-related distances to generate potential cluster centers, ensuring that peaks across different density regions are considered simultaneously. Furthermore, an MLP, guided by pseudo-labels from sub-clusters, is designed to learn low-dimensional embeddings for high-dimensional data, preserving data locality while enhancing clusterability. Extensive experiments demonstrate the effectiveness and superiority of EDPC against state-of-the-art DPC methods.

Multi-view clustering aims to integrate complementary information from multiple views to improve clustering performance. However, existing ensemble-based methods suffer from information loss due to their reliance on single-granularity labels, limiting the discriminative capability of learned representations. Meanwhile, representation and graph fusion-based approaches face challenges such as explicit view alignment and manual weight tuning, making them less effective for heterogeneous views with varying data distributions. To address these limitations, we propose a novel multi-view clustering framework via Multi-granularity Ensemble (MGE), fully using the multi-granularity information across diverse views for accurate and consistent clustering. Specifically, MGE first modifies the hierarchical clustering and then leverages it on each view (including the fused view) to achieve multi-granularity labels. Moreover, the cross-view and cross-granularity fusion strategy is designed to learn a robust co-association similarity matrix, which effectively preserves the fine-grained and coarse-grained structures of multi-view data and facilitates subsequent clustering. Therefore, MGE can provide a comprehensive representation of local and global patterns within data, eliminating the requirement for view alignment and weight tuning. Experiments demonstrate that MGE consistently outperforms state-of-the-art methods across multiple datasets, validating its effectiveness and superiority in handling heterogeneous views.

Density-based clustering algorithms are widely used for their ability to handle complex datasets; however, their performance often depends on the definition of density and is sensitive to data shapes, density variations, and noise. To address these challenges, we propose a novel parameter-free clustering algorithm named "Bombing", which can automatically determine the number of clusters. The algorithm enhances adaptability to complex data structures by combining global and local density estimations and utilizing dynamically adjusted density measures. By introducing the concept of generalized clusters and employing a "bombing" process that propagates clustering outward from core points, the method effectively mitigates the impact of cluster shapes, overlaps, and noise. Additionally, through the analysis of adjacency degrees, the algorithm can automatically detect the optimal number of clusters without prior knowledge. Experimental results on various synthetic and real datasets show that our proposed algorithm outperforms existing methods in clustering accuracy and robustness. It effectively handles noise and cluster overlap, and excels in identifying the optimal value of K .

Clustering is an essential analytical tool across a wide range of scientific fields, including biology, chemistry, astronomy, and pattern recognition. This paper introduces a novel clustering algorithm, called Torque Clustering, as a competitive alternative to existing methods, based on the intuitive principle that a cluster should merge with its nearest neighbor with a higher mass, unless both clusters have relatively large masses and the distance between them is also substantial. By identifying peaks in mass and distance, the algorithm effectively detects and removes incorrect mergers. The proposed method is entirely parameter-free, enabling it to autonomously recognize various cluster types, determine the optimal number of clusters, and identify noise. Extensive experiments on synthetic and real-world data sets demonstrate the algorithm's versatility and consistently strong performance compared to other state-of-the-art methods.

Protein-protein interactions (PPIs) are fundamental to understanding cellular mechanisms, signaling networks, disease pathways, and drug development. Over the years, numerous computational models with artificial intelligence (AI) have been developed to predict PPIs. However, these models mostly face significant challenges, such as fragmented feature extraction pipelines, inability to capture complex global relationships among proteins, and reliance on handcrafted features. These challenges often limit their prediction accuracy. To address these issues, the Knowledge Graph Fused Graph Neural Network (KGF-GNN) was proposed, offering an end-to-end learning approach that integrates Protein Associated Network (PAN) with observed PPI data. While KGF-GNN achieves notable performance improvements, it focuses primarily on local topological features extracted by Graph Neural Networks (GNNs), potentially overlooking critical global patterns. Moreover, its feature fusion process lacks the flexibility to effectively combine diverse biological information. To overcome these shortcomings, this paper introduces a Hybrid Ensemble End-to-End Neural Network (HEENN), which incorporates three key innovations: (1) Local Feature Extraction via Graph Attention Network (GAT): HEENN employs GAT to enable more precise extraction of local topological and semantic features, allowing the model to focus on the most relevant interactions and relationships within the data. (2) Global Feature Extraction via AutoEncoder: By leveraging an AutoEncoder framework, HEENN captures comprehensive global features from PANs and PPI datasets, complementing the GAT's local features to produce richer protein representations. (3) Attention-Enhanced Feature Fusion: An attention mechanism is employed during feature fusion to ensure an adaptive and effective integration of local and global features. Extensive experiments on real-world PPI datasets demonstrate that HEENN significantly outperforms KGF-GNN and other state-of-the-art models, achieving superior accuracy in PPI prediction. These advancements underscore the potential of HEENN in AI-driven bioinformatics research, which offers new opportunities for biological discovery and therapeutic innovation.

Granular-ball computing (GBC) proposed by Xia adaptively generates a different neighborhood for each object, resulting in greater generality and flexibility. Moreover, GBC greatly improves the efficiency by replacing point input with granular-ball. However, the current GB-based classifiers rigidly assign a specific class label to each data instance and lacks of the necessary strategies to address uncertain instances. These far-fetched certain classification approachs toward uncertain instances may suffer considerable risks. In this article, we introduce three-way decision (3WD) into GBC to construct a novel three-way decision with fuzzy granular-ball rough sets (3WD-FGBRS) from the perspective of uncertainty. This helps to construct reasonable multigranularity spaces for handling complex decision problems with uncertainty. First, 3WD-FGBRS is constructed in a data-driven method based on fuzziness, which avoids the subjective definition of certain risk parameters when calculating the threshold pairs. Based on 3WD-FGBRS, we further propose a sequential three-way decision with fuzzy granular-ball rough sets (S3WD-FGBRS) and analyze the fuzziness loss of multilevel decision result in S3WD-FGBRS. Then, the optimal granular-ball space selection mechanism of S3WD-FGBRS is introduced by combining fuzziness and granular-ball space distance. Finally, extensive comparative experiments are conducted with 3 state-of-the-art GB-based classifiers and 3 classical machine learning classifiers on 12 public benchmark datasets. The results show that our models almost outperform other comparison methods in terms of effectiveness, efficiency and robustness.

As an innovative methodology in data processing and knowledge representation, granular-ball computing (GBC) adaptively generates distinct neighborhoods for individual objects, thereby improving both generality and flexibility. By replacing point inputs with granular-balls (GBs), GBC achieves substantial efficiency gains. However, traditional GB-based classifiers may produce unreliable classifications under uncertain conditions. To address this limitation, we propose a novel approach that integrates three-way decision (3WD) theory with GBC, enabling robust handling of uncertain classification problems. This study first introduces a sequential three-way decision with fuzzy granular-ball rough sets (S3WD-FGBRS). We systematically analyze the changing rules of the multilevel decision cost in S3WD-FGBRS and its three regions. Building upon the principle of justifiable granularity, we develop a cost-sensitive three-way granular-ball generation method (CS3W-GBG) based on S3WD-FGBRS that incorporates a granularity optimization mechanism. To validate our approach, we conduct comprehensive experiments using three state-of-the-art GB classifiers and two benchmark classifiers on 12 publicly available datasets. Experimental results demonstrate that CS3W-GBG exhibits strong resilience in processing uncertain data through its 3WD strategy. Furthermore, our method achieves competitive performance compared to existing approaches in terms of classification accuracy and robustness.

Multi-label classification represented by hierarchical classification (HC) plays an important role in current large-scale problems, which can acquire a more accurate expression of data that conforms to the human multi-granularity cognitive process. To compress the original dataset and simultaneously enhance the expressive force of models, selecting an appropriate granularity for approximately describing the classification is the main task in the rough set theory. Nevertheless, the current rough set theory merely concerns flat classification and encounters new problems when approximately describing HC. 1) There lacks a measure to correctly reflect misclassification in accordance with the hierarchical accuracy of HC on the training set. 2) There lacks a measure relying on the distribution of the dataset to reflect the difference between two distinct feature sets describing HC in generalization ability. To address the mentioned issues, this paper utilizes the knowledge distance to characterize HC and proposes a cost-sensitive granularity selection for HC. First, HC and features are respectively granulated according to hierarchical quotient space and neighborhood granular structures. Then, knowledge distance and its extended form are employed to formulate misclassification and test costs. On this basis, a cost-sensitive neighborhood granularity selection is presented for HC. Finally, we experimentally demonstrate the excellent performance of the proposed method in terms of efficiency and HC accuracy both in synthetic and real datasets.

The granular-ball (GB)-based classifier introduced by Xia exhibits adaptability in creating coarse-grained information granules for input, thereby enhancing its generality and flexibility. Nevertheless, the current GB-based classifiers rigidly assign a specific class label to each data instance and lack the necessary strategies to address uncertain instances. These far-fetched certain classification approaches toward uncertain instances may suffer considerable risks. To solve this problem, we construct a robust three-way classifier with shadowed GBs (3WC-SGBs) for uncertain data. First, combined with information entropy, we propose an enhanced GB generation method with the principle of justifiable granularity. Subsequently, based on minimum uncertainty, a shadowed mapping is utilized to partition a GB into core region (COR), important region (IMP), and unessential region (UNE). Based on the constructed shadowed GBs, we establish a three-way classifier to categorize data instances into certain classes and uncertain case. Finally, extensive comparative experiments are conducted with two three-way classifiers, three state-of-the-art GB-based classifiers, and three classical machine learning classifiers on 12 public benchmark datasets. The results show that our model demonstrates robustness in managing uncertain data and effectively mitigates classification risks. Furthermore, our model almost outperforms the other comparison methods in both effectiveness and efficiency.

Protein-protein interactions (PPIs) are the foundation for numerous biological processes within cells, which are crucial for understanding cellular signaling networks, disease mechanisms, and drug development. Recently, numerous artificial intelligence (AI)-based approaches have emerged for predicting PPIs. Nevertheless, existing AI-based approaches either partially or loosely consider these relationships and mechanisms by a non-end-to-end learning framework, resulting in sub-optimal feature extractions and fusions for prediction. To address this issue, this paper proposes an end-to-end graph neural network model for protein-protein interaction prediction, termed the Knowledge Graph Fused Graph Neural Network (KGF-GNN). First, protein associated network (PAN) is constructed by comprehensively exploiting protein-associated relationships and mechanisms among drugs, diseases, ribonucleic acid, protein structures, etc. Then, a graph neural network (GNN) is built to extract both the topological and semantic features from PAN. Secondly, the observed interactions between proteins are constructed into a PPI network, and another GNN is built to extract the hidden topological features within the PPI network. Third, a multi-layer perceptron is designed to fuse the extracted various features by end-to-end learning. With such designs, the feature extractions and fusions of PPIs are guaranteed to be comprehensive and optimal for prediction. Finally, by conducting extensive experiments on real PPI datasets, we demonstrate that our KGF-GNN can accurately predict PPIs and significantly outperform state-of-the-art models.

Network data is ubiquitous, such as telecommunication, transport systems, online social networks, protein-protein interactions, etc. Since the huge scale and the complexity of network data, former machine learning system tried to understand network data arduously. On the other hand, thought of multi-granular cognitive computation simulates the problem-solving process of human brains. It simplifies the complex problems and solves problems from the easier to harder. Therefore, the application of multi-granularity problem-solving ideas or methods to deal with network data mining is increasingly adopted by researchers either intentionally or unintentionally. This paper looks into the domain of network representation learning (NRL). It systematically combs the research work in this field in recent years. In this paper, it is discovered that in dealing with the complexity of the network and pursuing the efficiency of computing resources, the multi-granularity solution becomes an excellent path that is hard to go around. Although there are several papers about survey of NRL, to our best knowledge, we are the first to survey the NRL from the perspective of multi-granular computing. This paper proposes the challenges that NRL meets. Furthermore, the feasibility of solving the challenges of NRL with multi-granular computing methodologies is analyzed and discussed. Some potential key scientific problems are sorted out and prospected in applying multi-granular computing for NRL research.

Protein-protein interactions (PPIs) are essential to understanding cellular mechanisms, signaling networks, disease processes, and drug development, as they represent the physical contacts and functional associations between proteins. Recent advances have witnessed the achievements of artificial intelligence (AI) methods aimed at predicting PPIs. However, these approaches often handle the intricate web of relationships and mechanisms among proteins, drugs, diseases, ribonucleic acid (RNA), and protein structures in a fragmented or superficial manner. This is typically due to the limitations of non-end-to-end learning frameworks, which can lead to sub-optimal feature extraction and fusion, thereby compromising the prediction accuracy. To address these deficiencies, this paper introduces a novel end-to-end learning model, the Knowledge Graph Fused Graph Neural Network (KGF-GNN). This model comprises three integral components: (1) Protein Associated Network (PAN) Construction: We begin by constructing a PAN that extensively captures the diverse relationships and mechanisms linking proteins with drugs, diseases, RNA, and protein structures. (2) Graph Neural Network for Feature Extraction: A Graph Neural Network (GNN) is then employed to distill both topological and semantic features from the PAN, alongside another GNN designed to extract topological features directly from observed PPI networks. (3) Multi-layer Perceptron for Feature Fusion: Finally, a multi-layer perceptron integrates these varied features through end-to-end learning, ensuring that the feature extraction and fusion processes are both comprehensive and optimized for PPI prediction. Extensive experiments conducted on real-world PPI datasets validate the effectiveness of our proposed KGF-GNN approach, which not only achieves high accuracy in predicting PPIs but also significantly surpasses existing state-of-the-art models. This work not only enhances our ability to predict PPIs with a higher precision but also contributes to the broader application of AI in Bioinformatics, offering profound implications for biological research and therapeutic development.

In large-scale decision systems with high dimensions, constructing an efficient feature selection method via an uncertainty measure, has become a critical problem in fuzzy rough sets (FRS). However, the uncertainty method constructed through FRS for feature selection has the following limitations. 1) The composition of the uncertainty caused by fuzzy distance and similarity is neglected, which can not precisely evaluate the uncertainty. 2) The method of measuring uncertainty is to select all the sample for establishing a fuzzy similarity matrix, which leads to substantial time consumption. 3) The efficiency of selecting import features in a nonbatch manner is relatively low. Driven by this, both granular-ball (GB) computing and three-way approximations (TWA) are integrated to guide an uncertainty measure named multigranularity fuzzy entropy (MGFE), which is based on fuzzy distance and similarity, to improve the efficiency of feature selection. The MGFE is primarily recommended for measuring the uncertainty in multigranularity spaces. Therewith, the GB and TWA computing are integrated to compress the sample space to select representative sample. In addition, the MGFE is employed to assess the significance of the features in the representative sample space. Aided by the TWA, an efficient filter-wrapper feature selection with a three-way accelerator is successfully developed. Finally, related experiments illustrate the advancement of our proposed feature selection.s

Three-way decision with neighborhood rough sets (3WDNRS) is adept at addressing uncertain problems involving continuous data by configuring the neighborhood radius. However, on one hand, the inputs of 3WDNRS are individual neighborhood granules, which reduce the decision efficiency and generality; on other hand, the thresholds of 3WDNRS require prior knowledge to be approximately set in advance, making it difficult to apply in cases where such knowledge is unavailable. To address these issues, we introduce granular-ball computing (GBC) into 3WDNRS from the perspective of uncertainty. Firstly, we propose an enhanced granular-ball generation method based on DBSCAN called DBGBC. Subsequently, we present an improved granular-ball neighborhood rough sets model (GBNRS++) by combining DBGBC with a quality index. Furthermore, we construct a three-way classifier with granular-ball neighborhood rough sets (3WC-GBNRS++) based on the principle of minimum fuzziness loss. This approach provides an objective and efficient way to determine the thresholds. To further enhance classification accuracy, we design an adaptive granular-ball neighborhood within the subsequent classification process of 3WC-GBNRS++. Finally, experimental results demonstrate that, 3WC-GBNRS++ almost outperformed other comparison methods in terms of effectiveness and robustness, including 4 state-of-the-art granular-balls-based classifiers and 5 classical machine learning classifiers on 12 public benchmark datasets. Moreover, we discuss the limitations of our work and the outlook for future research.

Currently, three-way decision with neighborhood rough sets (3WDNRS) is widely used in many fields. The core of 3WDNRS is to calculate threshold pairs to divide a neighborhood space into three pair-wise disjoint regions. The majority of research on 3WDNRS mainly aims to calculate thresholds with the given risk parameters to minimize the misclassification cost. However, in practical applications, risk parameters are often subjectively determined based on expert experience. This makes it challenging to accurately obtain the thresholds in 3WDNRS. To solve this problem, fuzziness is introduced into 3WDNRS to provide a new perspective on 3WD theory. First, a shadowed set framework is constructed, named three-way approximations based on shadowed sets (3WA-SS). Based on 3WA-SS, a datadriven adapted neighborhood (DAN) is constructed. Then, an improved fuzziness-based 3WDNRS (F′ -3WDNRS) is further proposed and optimized by minimizing uncertainty change to obtain more reasonable threshold pair based on DAN. Finally, extensive experiments are conducted on our proposed model, and the results show that F′ -3WDNRS is effective and reliable for making decisions.

Abstract This work presents a multi‐focus image fusion method based on Transformer and U‐Net with an unsupervised training fashion. In this work, the authors introduce Transformer into image fusion because it has great ability to capture the global dependencies and low‐frequency features. In image processing, convolutional neural network (CNN) has good performance of detailed feature extraction but a weakness for global feature extraction, and Transformer has limited power in local or detailed information extraction but a strong capacity for global feature extraction. Thus, this work combines the advantages of CNN and Transformer to propose an unsupervised decision map making model for image fusion joint U‐Net. The authors construct a model including feature extraction and feature reconstruction modules which correspond to the encoder and decoder network of U‐Net, respectively. In addition, perceptual loss is introduced on the basis of structural similarity loss because the combination of these two loss functions can achieve better performance with lower training cost. Experiments show that the proposed image fusion method performs better fusion performance compared with the existing methods.

Feature selection method with rough sets based on incremental learning has the major advantage of the higher efficiency in a dynamic information system, which has attracted extensive research. However, the incremental approximation feature selection with an accelerator (IAFSA) remains ambiguous for a dynamic information system with fuzzy decisions (ISFD). Driven by this concern, the nonincremental approximation feature selection is first presented by fuzzy knowledge distance (FKD). Second, the incremental theory of FKD is constructed with a batch of objects appended to or removed from the dynamic ISFD. Subsequently, an acceleration mechanism to eliminate redundant information granules is developed to reduce the sample space. Eventually, two categories of IAFSA based on FKD are presented. The experiments reflect the efficiency and effectiveness of the developed IAFSA algorithms.

Most existing multi-view clustering methods have problems with parameter selection and high computational complexity, and there have been very few works based on hierarchical clustering to learn the complementary information of multiple views. In this paper, we propose a Multi-view Adjacency-constrained Nearest Neighbor Clustering (MANNC) and its parameter-free version (MANNC-PF) to overcome these limitations. Experiments tested on eight real-world datasets validate the superiority of the proposed methods compared with the 13 current state-of-the-art methods.

Multi-focus image fusion can overcome the issues that optical lens imaging cannot focus multiple targets simultaneously due to the depth of field limitation. In this paper, we propose a generative adversarial network (DDF-GAN) which consists of a generator and two discriminators to directly generate fused images without decision maps and post-processing. In training process, the source all-in-focus image and the fused image generated by generator are used as input to one of the dual discriminators. Meanwhile, the gradient map of the fused image and the source gradient map of the source all-in-focus image are used as input to another discriminator. An adversarial relationship is established to enhance texture details of fused image. In addition, we create a data set and use it as the main training set for the proposed model. Abundant experiments were carried out to verify the availability of our method. Experimental results prove that our method has advantages in subjective visual perception of human and quantitative measurement.

To effectively improve the convergence and diversity of the multiobjective particle swarm optimization (MOPSO), we proposed a novel MOPSO based on ranking and cyclic distance (RCDMOPSO) that comprehensively considers the spatial target and congestion information of particles. RCDMOPSO introduced a method namely global proportional ranking (GPR) which differs from nondominated ranking under the Pareto framework, and designed a novel external archive maintenance and the global selection strategies of learning sample by combining GPR with cyclic distance. In this paper, RCDMOPSO together with eight classic and state-of-the-art algorithms were examined on ZDT, UF, and DTLZ series to test functions. The results show that RCDMOPSO is highly competitive in achieving the objectives of both convergence and diversity. RCDMOPSO outperformed other popular algorithms such as MOPSOs and multiobjective genetic algorithms based on comprehensive performance evaluation indicators inverted generational distance and hypervolume, thus supporting that RCDMOPSO is an effective approach to tackle multiobjective optimization problems.

Feature selection is an important preprocessing step for high-dimensional data mining and machine learning; it is viewed as the selection of the optimal granularity to describe the target concept in rough set theory. Currently, research on rough sets mainly focuses on granularity selection in flat classification scenarios, while organizing hundreds of labels for hierarchical classification (HC) can provide additional external information and achieve better performance in terms of both accuracy and efficiency. However, HC also faces the following problems: 1) the current measures’ failure to characterize the uncertainty in HC; 2) the inability to select the optimal granularity of the target concept in HC; and 3) no valid approach to select features in a decision system with hierarchical classification (HieDS). To address these problems, this article introduces HC into rough set theory and proposes an approach to granularity selection for HC. First, we introduce the knowledge distance to reflect the uncertainty of HC and define related important characteristic functions to describe a HieDS. Then, from the perspective of uncertainty, granularity selection for the target concept and feature selection are presented based on these characteristic functions. Finally, we experimentally realize granularity selection and demonstrate excellent performance of feature selection in a HieDS in terms of both feature selection and classification accuracy.

Strongly connected components (SCCs) are an important kind of subgraphs in digraphs. It can be viewed as a kind of knowledge in the viewpoint of knowledge discovery. In our previous work, a knowledge discovery algorithm called RSCC was proposed for finding SCCs of simple digraphs based on two operators, k-step R-related set and k-step upper approximation, of rough set theory (RST). RSCC algorithm can find SCCs more efficiently than Tarjan algorithm of linear complexity. However, on the one hand, as the theoretical basis of RST applied to SCCs discovery of digraphs, the theoretical relationships between RST and graph theory investigated in previous work only include four equivalences between fundamental RST and graph concepts relating to SCCs. The reasonability of using the two RST operators to find SCCs still need to be investigated. On the other hand, it is found that there are three SCCs correlations between vertices after we use three RST concepts, R-related set, lower and upper approximation sets, to analyze SCCs. RSCC algorithm ignores these SCCs correlations so that the efficiency of RSCC is affected negatively. For the above two issues, firstly, we explore the equivalence between the two RST operators and Breadth-First Search (BFS) which is one of the most basic graph search algorithms and the most direct way to find SCCs. These equivalences explain the reasonability of using the two RST operators to find SCCs, and enrich the content of the theoretical relationships between RST and graph theory. Secondly, we design a granulation strategy according to these three SCCs correlations. Then an algorithm called GRSCC for finding SCCs of simple digraphs based on granulation strategy is proposed. Experimental results show that GRSCC provides better performance to RSCC.

Different rough approximation spaces could be induced for an information system by its different attribute subsets, thus the multigranularity rough approximations of a fuzzy concept could be developed. Research on the uncertainty in multi-granulation spaces becomes a basic issue of uncertainty measure. If the uncertainty measure is not accurate enough, two different rough approximation spaces of a fuzzy concept may have the same uncertainty, and the difference between them for describing a fuzzy concept cannot be reflected. In this case, attribute reduction, granularity selection, and multigranularity measure cannot be conducted effectively. Therefore, establishing an uncertainty measure model with strong distinguishing ability in multi-granulation spaces is a key issue in uncertainty knowledge processing. In this paper, this problem will be solved in the view of knowledge distance. First, a fuzzy knowledge distance measure (FKD) based on the Earth Mover's distance is introduced. Even if two rough approximation spaces possess the same uncertainty when describing a fuzzy concept, they can be discriminated by FKD. Then, by studying the change rules of the FKD in a hierarchical quotient space structure, it is found that the FKD between any two rough approximation spaces in an HQSS is equal to the difference between their granularity measure or information measure. Furthermore, in order to show the applicability of the FKD, the FKD is used in granularity selection, attribute reduct, and multigranularity measure. The experimental results show that the FKD-based attribute significance function has a more powerful ability to obtain shorter reduct and it is more robustness, which show the effectiveness of the FKD.

Shadowed sets, proposed by Pedrycz, provide a threeway approximation scheme for transforming the universe of a fuzzy set into three disjoint areas, i.e., elevated area, reduced area, and shadow area. To calculate a pair of decision-making thresholds, an analytic method was proposed by solving a minimization problem of the uncertainty arising from the three areas. However, some uncertainty will be lost in the process of constructing the shadowed set model using Pedrycz's method. Moreover, few references on how to measure the uncertainty of shadow sets exist. In this article, a comprehensible method for measuring the fuzzy entropy of a shadowed set, i.e., interval fuzzy entropy, is defined. Based on the interval fuzzy entropy, a new shadowed set model, namely, interval shadowed sets, is proposed. Compared with Pedrycz's model, the main difference is that the range of the shadow area in this model is [β, α] (0 ≤ β <; α ≤ 1) while not [0, 1]. By solving a fuzzy entropy loss-minimization problem, a pair of optimal thresholds, α and β, can be obtained. Finally, the results of the instance analysis of different types of representative membership functions and many experiments show that the fuzzy entropy loss of the interval shadowed set is lower than that of the traditional shadowed set of a fuzzy set. These results enrich shadowed set theory from a new perspective.

The traditional [Formula: see text]-means algorithm has been widely used as a simple and efficient clustering method. However, the performance of this algorithm is highly dependent on the selection of initial cluster centers. Therefore, the method adopted for choosing initial cluster centers is extremely important. In this letter, we redefine the density of points according to the number of its neighbors, as well as the distance between points and their neighbors. In addition, we define a new distance measure that considers both Euclidean distance and density. Based on that, we propose an algorithm for selecting initial cluster centers that can dynamically adjust the weighting parameter. Furthermore, we propose a new internal clustering validation measure, the clustering validation index based on the neighbors (CVN), which can be exploited to select the optimal result among multiple clustering results. Experimental results show that the proposed algorithm outperforms existing initialization methods on real-world data sets and demonstrates the adaptability of the proposed algorithm to data sets with various characteristics.