With the growing demand for face editing applications, face inpainting has become an increasingly important subfield within image inpainting research. While many existing methods use semantic segmentation guidance, they apply uniform weighting across all regions of the map. Since the missing areas of the image lack meaningful features, this uniform treatment provides insufficient guidance in missing areas, leading to unrealistic or structurally incoherent results. Moreover, these methods generally lack the capacity to adapt inpainting results to individual user preferences, thereby limiting their effectiveness in personalized face editing. To address these limitations, we propose SegPainter, a Mamba-based architecture for user-controllable face inpainting that enables customized restoration guided by user-defined semantic segmentation maps generated using the Image Segmentation Annotation Tool (ISAT) integrated with Meta's Segment Anything Model (SAM). Specifically, we propose Hard Mask Soft One-Hot Encoding (HMSOE) to adaptively weight regions in the segmentation map based on whether they correspond to known or missing areas of the masked image. This strategy amplifies semantic guidance in missing regions while attenuating it in known regions to avoid over-constraining existing content. We further introduce Semantic-Guided State Space Model (SG-SSM) to dynamically modulate the Mamba layer with semantic features, adapting guidance to the masked image. To enhance the quality of inpainting results, we also propose Tri-Scan Inspection (TSI), a scanning mechanism designed to capture both global and local dependencies while preserving spatial continuity and facial structure. Extensive experiments on the CelebAMask-HQ and FFHQ datasets demonstrate that our framework outperforms state-of-the-art methods, producing sharper and more semantically consistent face inpainting results.

Existing optimization algorithms are designed based on the finest granularity, i.e., a point. This design limits global search capability and reduces efficiency in global exploration. To solve this problem, we propose a multi-granularity optimization algorithm (MGO), a novel evolutionary algorithm that uses granular-balls instead of points for optimization. MGO distributes a large number of granular-balls across the solution space to explore both critical regions, where optimal solutions are likely located, and non-critical regions for a broader search. Small, fine-granularity granular-balls focus on critical regions, while larger, coarse-granularity ones cover less critical areas. Additionally, there is competition and cooperation among different granular-balls to better utilize computational resources. This multi-granularity representation enhances global search efficiency and accelerates convergence. MGO outperforms many popular state-of-the-art algorithms in the CEC2013 single-objective optimization benchmarks and shows clear advantages in the social network influence maximization (IM) problem. With strong approximation capability, fast convergence speed, and simple design, MGO presents a competitive alternative to most existing optimization algorithms.

The Influence Maximization (IM) problem in social networks has been extensively studied, with greedy algorithms providing accurate and reliable solutions. However, their high computational cost renders them impractical for large-scale networks. Conversely, structure-based heuristic methods reduce computational complexity but often yield suboptimal solutions. To address these challenges, this paper proposes a novel cooperative and competitive multi-granularity fireworks algorithm (CCFWA). Unlike traditional point-based optimization algorithms, CCFWA employs spherical fireworks of varying granularity to enhance the search process. By distributing fireworks of different scales throughout the solution space, the algorithm effectively distinguishes between critical and less significant regions. Fine-granularity fireworks conduct detailed searches in key areas, while coarse-granularity fireworks enable rapid exploration of less important regions. This multi-granularity representation balances global exploration and local exploitation, improving both accuracy and efficiency in solving the IM problem. Experiments on four real-world social networks demonstrate that CCFWA consistently outperforms six state-of-the-art algorithms in terms of computational efficiency and solution quality.

Understanding the effect of hyperparameters of the network structure on the performance of Convolutional Neural Networks (CNNs) remains the most fundamental and urgent issue in deep learning, and we attempt to address this issue based on the piecewise linear (PWL) function nature of CNNs in this paper. Firstly, the operations of convolutions, ReLUs and Max pooling in a CNN are represented as the multiplication of multiple matrices for a fixed sample in order to obtain an algebraic expression of CNNs, this expression clearly suggests that CNNs are PWL functions. Although such representation has high time complexity, it provides a more convenient and intuitive way to study the mathematical properties of CNNs. Secondly, we develop a tight bound of the number of linear regions and the upper bounds of generalization error for CNNs, both taking into account factors such as the number of layers, dimension of pooling, and the width in the network. The above research results provide a possible guidance for designing and training CNNs.

In multilabel learning, the precise mining and appropriate application of label correlation can improve the effectiveness and generalization of prediction models. In order to characterize label correlation more carefully, the concept of operator-valued kernel is introduced. The value of operator-valued kernel is an operator on Hilbert space, and when applied to a practical problem, the function-valued operator degenerates into a positive-definite matrix, which aims to describe the label correlation. However, existing works focus on the basic theory of operator-valued kernel, and lack way to learn specific kernel from specific datasets, thus the application of operator-valued kernel in practical problem is greatly hindered. In this article, we focus on learning operator-valued kernels with fuzzy rough sets from multilabel datasets and designing learning algorithm for multilabel classification. First, the importance distribution of feature set to different labels at each sample is measured by using kernelized fuzzy rough sets. For a single sample, label correlation matrix is constructed based on the consistency of the importance distribution of features to labels, so as to characterize the correlation information between different labels. By considering the interaction information between two label correlation matrices, the label incidence matrix between two samples is obtained. Therefore, a new operator-valued kernel is defined by using label incidence matrices as elements. This operator-valued kernel is further proved to be an entangled and transformable kernel. On the basis, the proposed operator-valued kernel is applied to develop an efficient learning algorithm for multilabel classification. The generalization error bound of the prediction function is measured by Rademacher complexity. In order to illustrate the effectiveness of our algorithm, the classification experiments and statistical analysis results on twelve multilabel datasets are provided, which are compared with seven high performance algorithms.

Fuzzy rough sets (FRSs) theory is an important granular computing method to deal with incomplete information systems, and the attribute reduction is a basic key issue in FRSs. In this article, we construct a novel framework for selecting the optimal reduct of FRSs with theoretical guarantees by considering the influence of granule size and incorporating the stability theory in machine learning. First, a granule-based soft-margin support vector machine (GSSVM) algorithm is proposed for classification tasks by introducing the$\lambda$-conditional entropy into the hinge loss function, which takes the impact of granule size on data loss into account. Then, according to stability theory, the generalization error bound of the GSSVM algorithm is derived as a theoretical guarantee for selecting the optimal reduct. Finally, an optimization attribute reduction algorithm (RDROAR) based on the relative discernibility relation is presented by removing the attributes with low importance in a reduct while ensuring the generalization ability of GSSVM. Numerical experiments prove the effectiveness of the improved algorithm as well as verify the rationality and effectiveness of the optimal reduct.

Recently, many post hoc graph neural network (GNN) explanation methods have been explored to uncover GNNs' predictive behaviors by analyzing the embeddings produced by the GNN models. However, these methods suffer from explanation ambiguity inherent in learned graph embeddings because aggregation-based embeddings can lead to the loss of unique identifiers for individual graph components and, thus, allow noncausal nodes that are adjacent to true causal patterns to unintentionally embody causal information in their embeddings, hindering the explanations from faithfully representing the true insights of GNNs' predictive reasoning. In this article, we present an embedding ambiguity-sensitive GNN explanation framework (EAGX). EAGX can effectively mitigate the impact of embedding-induced explanation ambiguity by creating edges' ambiguity feature extractor, exploring edges' predictive relevance, and integrating them into the explanation process, thereby capturing each graph component's contribution to the predictions. Specifically, we first propose a centroid-constrained fuzzy c-means algorithm to construct an ambiguity feature extractor. Then, we leverage the ambiguity features for edges to develop the ambiguity-based edge attribution module for assigning a prediction relevance score to each edge. Finally, instead of focusing only on the edges with high influence to the GNN prediction, we introduce a joint optimization strategy to refine the learning process of our edge attribution module, empowering EAGX to capture the subtle interplay of both causal and noncausal subgraphs on model predictions, which further improve the explainability of GNN predictions. Experimental results demonstrate that EAGX outperforms the leading explainers on most evaluation metrics, underscoring its effectiveness in generating reliable and precise explanations for GNNs.

In multilabel learning, research on label correlation provides an effective solution to compress the hypothesis space of classifiers. However, this article focus on the label correlation adapted to overall data, while ignoring the locally targeted information presented by some instances. The lack exploration on the distribution of local label correlation in multilabel instance space undoubtedly limits the in-depth application of label correlation in multilabel learning. Based on the fuzzy rough set theory, the instance-level label correlation distribution is first proposed in this article and applied to design a novel multilabel learner. For each multilabel instance, the local importance of features to label is quantitatively analyzed, by considering the decisive influence of input information on decision making. According to coincidence degree between local feature weight distribution for different labels, the instance-level label correlation is constructed. In order to reflect the internal relationship between label variables objectively, the instance-level label correlation distribution is integrated into the empirical label relevance. On the basis, the label relevance matrix is used to define the constraints of the optimization function in a new form. The relative position of subseparating hyperplanes in input space is quantitatively characterized to reduce the complexity of the multilabel classifier and improve the learning performance. The experiment results on 18 multilabel datasets illustrate the effectiveness of our algorithm. The impact of core parameters on the performance is also dissected.

Formal concept analysis (FCA) extracts interpretable rules by using implication relationships between concepts, which is an effective method for acquiring knowledge. In this article, a fuzzy decision rule-based online classification algorithm called OFRCA is designed by combining online learning theory and FCA in a fuzzy formal decision context. First, the original weight vector of all attributes is obtained by fusing the original deterministic decision rules. Second, incremental fuzzy decision rules are obtained according to the added objects. Then, for the incremental fuzzy decision rules, OFRCA updates the weight vector based on minimizing the loss function. At the end of the process, all the rules are fused into an attribute-weighted classifier with the final weight vector. In this article, we employ the regret to obtain a learning guarantee for OFRCA. The growth rate of OFRCA's regret converges to 0 as the number of iterations increases in the ideal state, which provides an effective learning guarantee for OFRCA. To verify the effectiveness of the proposed algorithm, this article conducted numerical experiments and systematically analyzed the experimental results, which demonstrated that OFRCA achieved the highest advanced classification performance among the compared algorithms.

Multioutput regression aims to predict multiple continuous outputs simultaneously using the common set of input variables. The significant challenge arises from modeling relevance between inputs and outputs. Moreover, the shortage of labeled multioutput data and the divergence of data are other factors that impede the development of multioutput regression problems. The recent emergence of transfer learning techniques, which have the ability of leveraging previously acquired knowledge from a similar domain, provide a solution to the above issues. In this article, a novel fuzzy transfer learning method is proposed to tackle the multioutput regression problems in homogeneous and heterogeneous scenarios. By considering output–input dependencies and inter-output correlations, fuzzy rules are extracted to reflect the shared characteristics of different outputs and capture their uniqueness. For a homogeneous scenario, fuzzy rules are first accumulated in a related domain (called the source domain), which has a sufficient amount of training data. Based on different transform strategies, the fuzzy rules are then transferred to improve the new but similar regression tasks in the current domain (called the target domain), where only a few data have multiple responses. On this basis, we handle a more complex heterogeneous scenario by learning a latent input space to reduce the disagreement of variables between domains. The experiment results on thirteen real-world datasets with multiple outputs illustrate the effectiveness of our method. The impact of core coefficients on performance is also analyzed.

Granularrules have been extensively used for classification in fuzzy datasets to promote the advancement of artificial intelligence. However, due to the diversity of data types, how to improve the readability of the extracted granular rules while ensuring efficiency is always a challenge. Since granular reduct in granular computing (GrC) can simplify real complex problem and dataset, this article carries out granular rule learning from the perspective of granular reduct by taking formal concept analysis (FCA)-based GrC method as a framework. Specifically, for achieving classification task, we first propose a method to update the granular reduct, and then explore the updating mechanism of fuzzy granular rule in a reduced dataset. Second, a novel fuzzy rule-based classification model named FRCM is presented for fuzzy granular rule learning. In order to verify the effectiveness of the proposed model, some numerical experiments for incremental learning and fuzzy rule mining are conducted to demonstrate that FRCM can achieve the state-of-the-art classification performance.

The concept of dependency in a neighborhood rough set model is an important evaluation function for the feature selection. This function considers only the classification information contained in the lower approximation of the decision while ignoring the upper approximation. In this paper, we construct a class of uncertainty measures: decision self-information for the feature selection. These measures take into account the uncertainty information in the lower and the upper approximations. The relationships between these measures and their properties are discussed in detail. It is proven that the fourth measure, called relative neighborhood self-information, is better for feature selection than the other measures, because not only does it consider both the lower and the upper approximations but also the change of its magnitude is largest with the variation of feature subsets. This helps to facilitate the selection of optimal feature subsets. Finally, a greedy algorithm for feature selection has been designed and a series of numerical experiments was carried out to verify the effectiveness of the proposed algorithm. The experimental results show that the proposed algorithm often chooses fewer features and improves the classification accuracy in most cases.

Classical rough set theory is considered a useful tool for dealing with the uncertainty of categorical data. The major deficiency of this model is that the classical rough set model is sensitive to noise in classification learning due to the stringent condition of equivalence relation. Thus, a class of fuzzy similarity relations was introduced to describe the similarity between samples with categorical attributes. However, these kinds of similarity relations also have deficiencies when they are used in fuzzy rough computation. In this article, we propose a new fuzzy-rough-set model for categorical data by introducing a variable parameter to control the similarity of samples. This model employs the iterative computation strategy to define fuzzy rough approximations and dependence functions. It is proved that the proposed rough dependence function is monotonic. Finally, the proposed model is applied to the attribute reduction of categorical data. The experimental results indicate that the proposed model is more effective for categorical data than some existing algorithms.

Feature selection is a popular technique of preprocessing data. In order to deal with dynamic or large data, incremental feature selection has been developed, in which the features selected from existing data are integrated with those mined from both existing and dynamic data in the manner of incremental computation. Fuzzy rough set theory is powerful in handling uncertainty in real-valued data or even mixed data, and one of its most important applications is feature selection. Nevertheless, not much work has been found on fuzzy-rough-set-based incremental feature selection. Therefore, in this article, we investigate the incremental feature selection using a fuzzy-rough-set-based information entropy with incoming instances. Specifically, the representative instances from the incoming ones are first selected according to the information coverage of fuzzy granules generated by fuzzy rough sets. Then, the incremental mechanism of the fuzzy-rough-set-based information entropy is formulated by adding newcome instances. Finally, an incremental feature selection procedure, which we call active incremental feature selection, is proposed. Furthermore, some numerical experiments are conducted to assess the performance of the proposed feature selection algorithm, and the results show that our algorithm is of a prominent advantage in terms of computational time, especially for a dataset with large number of instances.

Fuzzy similarity relation is a function to measure the similarity between two samples. It is widely used to learn knowledge under the framework of fuzzy machine learning. The selection of a suitable fuzzy similarity relation is important for the learning task. It has been pointed out that fuzzy similarity relations can be brought into the framework of kernel functions in machine learning. This fact motivates us to study fuzzy similarity relation selection for fuzzy machine learning utilizing kernel selection methods in machine learning. Kernel alignment is a kernel selection method that is effective and has low computational complexity. In this paper, we present novel methods for fuzzy similarity relation selection based on the kernel alignment, and their use in attribution reduction for heterogeneous data. First, we define an ideal kernel for classification problems, based on which a novel fuzzy kernel alignment model is proposed. Second, we present a method for the fuzzy similarity relation selection based on the minimization of the fuzzy alignment between the defined ideal kernel and a kernel for the learning problem at hand. In order to show the correctness of this selection method, we prove that the lower bound of the classification accuracy of a support vector machine will increase with the decrease of the fuzzy alignment value. Furthermore, we apply the proposed fuzzy similarity relation selection to attribute reduction for heterogeneous data. Finally, we present experimental results to show that the proposed method of fuzzy similarity relation selection based on the fuzzy kernel alignment is effective.

At present, there is no unified method for solving multiattribute decision-making problems. In this paper, we propose two methods that benefit from some novel fuzzy rough set models. Some theoretical preliminaries pave the way. First, by means of a fuzzy logical implicator Iand a triangular norm T, four types of coverings-based variable precision (I, T)-fuzzy rough set models are proposed. They can be used to deal with misclassification and perturbation (here, misclassification refers to error or missing values in classification, while perturbation refers to small changes in digital data). Second, the properties and the relationships among these models are investigated. Finally, we rely on their remarkable features in order to establish two approaches to multiattribute decision-making. Some numerical examples illustrate the application of these new approaches. The sensitivity and comparative analyses show that the respective ranking results produced by these decision-making methods have a high consensus for multiattribute decision-making problems with fuzzy evaluation information.

Feature selection based on fuzzy rough sets is an effective approach to select a compact feature subset that optimally predicts a given decision label. Despite being studied extensively, most existing methods of fuzzy rough set based feature selection are restricted to computing the whole dataset in batch, which is often costly or even intractable for large datasets. To improve the time efficiency, we investigate the incremental perspective for fuzzy rough set based feature selection assuming data can be presented in sample subsets one after another. The key challenge for the incremental perspective is how to add and delete features with the subsequent arrival of sample subsets. We tackle this challenge with strategies of adding and deleting features based on the relative discernibility relations that are updated as subsets arrive sequentially. Two incremental algorithms for fuzzy rough set based feature selection are designed based on the strategies. One updates the selected features as each sample subset arrives, and outputs the final feature subset where no sample subset is left. The other updates the relative discernibility relations but only performs feature selection where there is no further subset arriving. Experimental comparisons suggest our incremental algorithms expedite fuzzy rough set based feature selection without compromising performance.

Feature selection is viewed as an important preprocessing step for pattern recognition, machine learning, and data mining. Neighborhood is one of the most important concepts in classification learning and can be used to distinguish samples with different decisions. In this paper, a neighborhood discrimination index is proposed to characterize the distinguishing information of a neighborhood relation. It reflects the distinguishing ability of a feature subset. The proposed discrimination index is computed by considering the cardinality of a neighborhood relation rather than neighborhood similarity classes. Variants of the discrimination index, including joint discrimination index, conditional discrimination index, and mutual discrimination index, are introduced to compute the change of distinguishing information caused by the combination of multiple feature subsets. They have the similar properties as Shannon entropy and its variants. A parameter, named neighborhood radius, is introduced in these discrimination measures to address the analysis of real-valued data. Based on the proposed discrimination measures, the significance measure of a candidate feature is defined and a greedy forward algorithm for feature selection is designed. Data sets selected from public data sources are used to compare the proposed algorithm with existing algorithms. The experimental results confirm that the discrimination index-based algorithm yields superior performance compared to other classical algorithms.

A fuzzy rough set is an important rough set model used for feature selection. It uses the fuzzy rough dependency as a criterion for feature selection. However, this model can merely maintain a maximal dependency function. It does not fit a given dataset well and cannot ideally describe the differences in sample classification. Therefore, in this study, we introduce a new model for handling this problem. First, we define the fuzzy decision of a sample using the concept of fuzzy neighborhood. Then, a parameterized fuzzy relation is introduced to characterize the fuzzy information granules, using which the fuzzy lower and upper approximations of a decision are reconstructed and a new fuzzy rough set model is introduced. This can guarantee that the membership degree of a sample to its own category reaches the maximal value. Furthermore, this approach can fit a given dataset and effectively prevents samples from being misclassified. Finally, we define the significance measure of a candidate attribute and design a greedy forward algorithm for feature selection. Twelve datasets selected from public data sources are used to compare the proposed algorithm with certain existing algorithms, and the experimental results show that the proposed reduction algorithm is more effective than classical fuzzy rough sets, especially for those datasets for which different categories exhibit a large degree of overlap.

Attribute reduction with rough sets is an effective technique for obtaining a compact and informative attribute set from a given dataset. However, traditional algorithms have no explicit provision for handling dynamic datasets where data present themselves in successive samples. Incremental algorithms for attribute reduction with rough sets have been recently introduced to handle dynamic datasets with large samples, though they have high complexity in time and space. To address the time/space complexity issue of the algorithms, this paper presents a novel incremental algorithm for attribute reduction with rough sets based on the adoption of an active sample selection process and an insight into the attribute reduction process. This algorithm first decides whether each incoming sample is useful with respect to the current dataset by the active sample selection process. A useless sample is discarded while a useful sample is selected to update a reduct. At the arrival of a useful sample, the attribute reduction process is then employed to guide how to add and/or delete attributes in the current reduct. The two processes thus constitute the theoretical framework of our algorithm. The proposed algorithm is finally experimentally shown to be efficient in time and space.

Multi-output regression estimation aims at mining a vector-valued function from multi-dimensional input vector to multi-dimensional output vector. However, the output variables may be correlative. It is desirable to develop a multi-dimensional regression model taking advantage of the possible correlations. Therefore, this paper proposes a novel multi-output support vector regression model via double regularization. For each output variable, we first introduce an influential level vector with the dimensionality equal to the one of the input vector, in order to characterize the correlation between this variable and other output variables. As a second regularization term, 2-norms of all influential level vectors are then added into the objective function. Each influential level vector is also considered in constraints of our model. Finally, experimental comparisons demonstrate that our proposed model in this paper has a better generalization performance as well as a better robustness.

Attribute reduction with rough sets aims to delete superfluous condition attributes from a decision system by considering the inconsistency between condition attributes and the decision labels. However, heterogeneous condition attributes including symbolic and real-valued ones always coexist for most decision systems and different types of attributes induce different kinds of granular structures. The existing rough set models do not have explicit mechanisms to address different kinds of granular structures reasonably and effectively. In this paper, we aim to perform attribute reduction for decision systems with symbolic and real-valued condition attributes by composing classical rough set and fuzzy rough set models. We first define a discernibility relation for every symbolic and real-valued condition attribute to characterize its discernible ability related to the decision labels. With these discernibility relations, we can develop a dependence function to measure the inconsistency between heterogeneous condition attributes and decision labels, and attribute reduction aims to keep this dependence function with a small perturbation. The proposed attribute reduction deals with heterogeneous condition attributes from the viewpoint of discernible ability and can consider the mutual effects between two types of attributes without preprocessing into single-typed ones. An algorithm to find reducts is developed and experiments are performed to demonstrate that the proposed idea is effective.

Determining the informativeness of unlabeled samples is a key issue in active learning. One solution to this is using the sample's inconsistency between conditional features and decision labels. In this paper, a fuzzy-rough-set-based active learning model is proposed to tackle this problem. First, the consistence degree of a labeled sample is computed by the lower approximations in fuzzy rough set, which reflects its minimum membership in the decision class. Then, the concept of sample covering is proposed to measure the relationship between labeled samples and unlabeled samples. Afterward, the memberships of an unlabeled sample belonging to different decision classes are computed based on the covering degrees of labeled samples on it. Finally, these memberships are used to form a sample selection criterion to measure the sample's inconsistency. By applying Gaussian kernel-based similarity relation to the aforementioned processes, a support vector machine (SVM)-based active learning scheme is developed. Experimental results demonstrate the effectiveness of the proposed model.

Covering information systems and ordered information systems are two important types of information systems. In this paper the relationships between covering information systems and ordered information systems are first examined, and it is proved that these two types of information systems are isomorphic under given conditions and can be equivalently transformed into each other. Then, the approach to attribute reduction in ordered information systems is proposed. Based on the isomorphism and equivalence of transformation, the method of attribute reduction in a covering information system can be directly obtained according to the reduction approach in an ordered information system. A practical example is employed to show that the proposed method is an effective technique to deal with complex data sets.

Communication between information systems is a basic problem in granular computing, and the concept of homomorphism is a useful mathematical tool to study this problem. In this paper, some properties of communication between information systems based on fuzzy rough sets are investigated. The concepts of fuzzy relation mappings between universes are first proposed in order to construct a fuzzy relation of one universe according to the given fuzzy relation on the other universe. The main properties of the mappings are studied. The notions of homomorphism of information systems based on fuzzy rough sets are then proposed, and it is proved that properties of relation operations in the original information system and structural features of the system, such as approximations of arbitrary fuzzy sets and attribute reductions, are guaranteed in its image system under the condition of homomorphism.

Attribute reduction is one of the most meaningful research topics in the existing fuzzy rough sets, and the approach of discernibility matrix is the mathematical foundation of computing reducts. When computing reducts with discernibility matrix, we find that only the minimal elements in a discernibility matrix are sufficient and necessary. This fact motivates our idea in this paper to develop a novel algorithm to find reducts that are based on the minimal elements in the discernibility matrix. Relative discernibility relations of conditional attributes are defined and minimal elements in the fuzzy discernibility matrix are characterized by the relative discernibility relations. Then, the algorithms to compute minimal elements and reducts are developed in the framework of fuzzy rough sets. Experimental comparison shows that the proposed algorithms are effective.

Attribute reduction is the strongest and most characteristic result in rough set theory to distinguish itself to other theories. In the framework of rough set, an approach of discernibility matrix and function is the theoretical foundation of finding reducts. In this paper, sample pair selection with rough set is proposed in order to compress the discernibility function of a decision table so that only minimal elements in the discernibility matrix are employed to find reducts. First relative discernibility relation of condition attribute is defined, indispensable and dispensable condition attributes are characterized by their relative discernibility relations and key sample pair set is defined for every condition attribute. With the key sample pair sets, all the sample pair selections can be found. Algorithms of computing one sample pair selection and finding reducts are also developed; comparisons with other methods of finding reducts are performed with several experiments which imply sample pair selection is effective as preprocessing step to find reducts.

Driver fatigue detection based on computer vision is considered as one of the most hopeful applications of image recognition technology. The key issue is to extract and select useful features from the driver images. In this work, we use the propertie

Kernel machines and rough sets are two classes of commonly exploited learning techniques. Kernel machines enhance traditional learning algorithms by bringing opportunities to deal with nonlinear classification problems, rough sets introduce a human-focused way to deal with uncertainty in learning problems. Granulation and approximation play a pivotal role in rough sets-based learning and reasoning. However, a way how to effectively generate fuzzy granules from data has not been fully studied so far. In this study, we integrate kernel functions with fuzzy rough set models and propose two types of kernelized fuzzy rough sets. Kernel functions are employed to compute the fuzzy T-equivalence relations between samples, thus generating fuzzy information granules in the approximation space. Subsequently fuzzy granules are used to approximate the classification based on the concepts of fuzzy lower and upper approximations. Based on the models of kernelized fuzzy rough sets, we extend the measures existing in classical rough sets to evaluate the approximation quality and approximation abilities of the attributes. We discuss the relationship between these measures and feature evaluation function ReliefF, and augment the ReliefF algorithm to enhance the robustness of these proposed measures. Finally, we apply these measures to evaluate and select features for classification problems. The experimental results help quantify the performance of the KFRS.

In this paper, a new approach to improve least squares support vector machines is presented. We consider the membership of every sample in constraints, that is to say, every sample are not fully assigned to one class. The membership is computed by employing the technique of fuzzy rough sets, and then a new least squares support vector machine algorithm based on fuzzy rough sets is proposed, experiments are carried out to show that our idea in this paper is feasible and valid.

The fuzzy-rough set (FRS) methodology, as a useful tool to handle discernibility and fuzziness, has been widely studied. Some researchers studied on the rough approximation of fuzzy sets, while some others focused on studying one application of FRS: attribute reduction (i.e., feature selection). However, constructing classifier by using FRS, as another application of FRS, has been less studied. In this paper, we build a rule-based classifier by using one generalized FRS model after proposing a new concept named as ¿consistence degree¿ which is used as the critical value to keep the discernibility information invariant in the processing of rule induction. First, we generalized the existing FRS to a robust model with respect to misclassification and perturbation by incorporating one controlled threshold into knowledge representation of FRS. Second, we propose a concept named as ¿consistence degree¿ and by the strict mathematical reasoning, we show that this concept is reasonable as a critical value to reduce redundant attribute values in database. By employing this concept, we then design a discernibility vector to develop the algorithms of rule induction. The induced rule set can function as a classifier. Finally, the experimental results show that the proposed rule-based classifier is feasible and effective on noisy data.

The fuzzy rough set (FRS) model has been introduced to handle databases with real values. However, FRS was sensitive to misclassification and perturbation (here misclassification means error or missing values in classification, and perturbation means small changes of numerical data). The variable precision rough sets (VPRSs) model was introduced to handle databases with misclassification. However, it could not effectively handle the real-valued datasets. Now, it is valuable from theoretical and practical aspects to combine FRS and VPRS so that a powerful tool, which not only can handle numerical data but also is less sensitive to misclassification and perturbation, can be developed. In this paper, we set up a model named fuzzy VPRSs (FVPRSs) by combining FRS and VPRS with the goal of making FRS a special case. First, we study the knowledge representation ways of FRS and VPRS, and then, propose the set approximation operators of FVPRS. Second, we employ the discernibility matrix approach to investigate the structure of attribute reductions in FVPRS and develop an algorithm to find all reductions. Third, in order to overcome the NP-complete problem of finding all reductions, we develop some fast heuristic algorithms to obtain one near-optimal attribute reduction. Finally, we compare FVPRS with RS, FRS, and several flexible RS-based approaches with respect to misclassification and perturbation. The experimental comparisons show the feasibility and effectiveness of FVPRS.

Fuzzy rough sets are the generalization of traditional rough sets to deal with both fuzziness and vagueness in data. The existing researches on fuzzy rough sets are mainly concentrated on the construction of approximation operators. Less effort has been put on the attributes reduction of databases with fuzzy rough sets. This paper mainly focuses on the attributes reduction with fuzzy rough sets. After analyzing the previous works on attributes reduction with fuzzy rough sets, we introduce formal concepts of attributes reduction with fuzzy rough sets and completely study the structure of attributes reduction. An algorithm using discernibility matrix to compute all the attributes reductions is developed. Based on these lines of thought, we set up a solid mathematical foundation for attributes reduction with fuzzy rough sets. The experimental results show that the idea in this paper is feasible and valid.

In fuzzy rough sets a fuzzy T-similarity relation is employed to describe the similarity degree between two objects and to construct lower and upper approximations for arbitrary fuzzy sets. Different triangular norm T identifies different point of view of similarity. Thus a reasonable selection of triangular norm is meaningful to practical applications of fuzzy rough sets. In this paper we discuss the selection of triangular norm and emphasize the well-known Lukasiewicz's triangular norm TLas a reasonable selection. And then we focus on attributes reduction with TL-fuzzy rough sets. First we define attributes reduction with TL-fuzzy rough sets and characterize it by dependency function. Second, the structure of proposed attributes reduction is investigated in detail by the approach of discernibility matrix. Third, an algorithm of computing attributes reduction is designed and finally a comparison with other methods of attributes reduction with fuzzy rough sets is performed using several experiments. The experimental results show that the method of attributes reduction with TL-fuzzy rough sets is feasible and valid.

This paper mainly focuses on the attribute reduction in fuzzy rough sets. An algorithm using discernibility matrix to compute all the attribute reductions is developed. After reducing the attributes, we introduce Support Vector Machine (SVM) as a classification technique to test the knowledge representation ability of attribute reduction. The numerical results show that the attribute reduction with fuzzy rough sets contains the same information as the original one.

Fuzzy rough set is the generalization of Pawlak rough set to deal with both of fuzziness and vagueness in data. The existing research on fuzzy rough sets mainly concentrates on the construction of approximation operators. Less effort has been put on the attributes reduction of fuzzy rough sets. This paper mainly focuses on the attributes reduction of fuzzy rough sets. After analyzing the previous work of reduction of fuzzy rough sets, formal concepts of attributes reduction of fuzzy rough sets are introduced and the structure of reduction is studied completely. An algorithm based on discernibility matrix to compute all the attributes reductions is developed. These concepts have been demonstrated by an example.

Rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty, it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets. The existing generalizations of fuzzy rough sets are all based on special fuzzy relations (fuzzy similarity relations, T-similarity relations), it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches. In this paper, from the viewpoint of constructive approach, we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them, the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined. In axiomatic approach, we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. The lattice and topological structures of fuzzy rough sets are also proposed. In order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets, a special lower approximation operator is applied to a fuzzy reasoning system, which coincides with the Mamdani algorithm.