A Galois adjunction, as an extension of Galois theory, originated from the discussion of the problem of solving polynomial equations using radicals. A fuzzy rough set, as a generalization of classic rough sets, is an important tool for handling complex data. This paper investigates lattice structures induced by rough approximation operators, with side results on Galois adjunctions and attribute reduction. First, we show that a fuzzy rough operator pair is a Galois adjunction between two fuzzy power sets if and only if the fuzzy relation degenerates into a crisp relation. Second, for each serial relation, we endow the family of all definable fuzzy sets a complete lattice structure and the lattice can be induced by an equivalence relation. Finally, for relation systems, we study attribute reduction that keeps a lattice constant and obtain an algorithm to identify all reducts.
Multi-Attribute Decision-Making (MADM) often involves probabilistic linguistic uncertainty, conflicting attributes, and evolving decision preferences, which pose challenges to traditional aggregation-based methods. To address these issues, this study proposes a novel framework that integrates stochastic dominance tensors with a dynamic three-way decision mechanism. The proposed approach constructs a Total Dominance Tensor based on first-order and second-order stochastic dominance, enabling structured representation of pairwise dominance relationships while preserving the distributional characteristics of probabilistic linguistic term sets (PLTS). To enhance decision flexibility, a data-driven adaptive threshold mechanism is developed, which iteratively adjusts the boundaries of acceptance, rejection, and deferment regions through a feedback process based on global dominance patterns. Compared with conventional methods, the proposed framework avoids premature information aggregation and provides finer discrimination among alternatives by explicitly modeling both strong and weak dominance structures. A case study on smartphone selection, along with comparative analysis against representative methods, demonstrates that the proposed method achieves consistent ranking results while improving robustness and differentiation under uncertainty. The proposed framework offers a practical and interpretable solution for decision-making problems involving complex probabilistic linguistic information and dynamic evaluation environments.
Achieving collective consensus in large-scale group decision-making (LSGDM) is increasingly challenging due to the explosion of heterogeneous information sources. The core problem lies in distilling coherent outcomes from massive datasets characterized by sparse user ratings, uncertain social trust, and complex strategic interactions. Existing approaches often fragment this problem, treating trust modeling, data completion, and consensus negotiation as isolated processes, thereby failing to capture their intrinsic interdependence. To address this unified challenge, we propose a granular ball (GB) evolutionary game-based framework. Unlike traditional point-based models, we represent each decision maker as a center–radius GB, where the radius inherently encodes uncertainty and social reliability. We introduce an adaptive community detection strategy based on GB distances to reveal latent groups with high consensus potential without requiring prior structural knowledge. Furthermore, we design a two-stage dynamic consensus mechanism that combines payoff-driven intra-group opinion evolution with inter-group coordination. During consensus reaching, individual strategies, opinions, and trust weights interact dynamically, while the previously obtained community structure provides hierarchical guidance for the evolution process. Experimental results demonstrate that the proposed method achieves robust and scalable consensus formation under sparse and uncertain conditions, outperforming existing approaches in adaptability and solution quality. The source code is openly available at https://github.com/JustinaZhan/GBLSGDM.
The granular ball fuzzy rough set model integrates the dual theoretical frameworks of granular computing and fuzzy rough sets. Owing to its strong capabilities in knowledge representation and data processing, it has been widely adopted in data analysis and mining tasks. This paper critically examines the limitations of existing granular ball fuzzy rough set models. By employing a highly flexible multidimensional semi-overlap function as an information aggregation operator, we propose a novel granular ball fuzzy rough set model, termed GBSGFRS, and apply it to both feature selection and anomaly detection. Specifically, this paper firstly illustrates, through two concrete examples, the deficiencies of current models in capturing fuzzy similarity relationships among data objects. It then introduces a method for constructing fuzzy similarity relations under multi-attribute conditions, the semi-overlap function method. Secondly, we formally define the GBSGFRS model by synergistically integrating granular balls with multidimensional semi-overlap functions. Based on this model, we derive a degree based reduction oriented feature selection algorithm, named GBSG-RD. Extensive experiments on 12 benchmark datasets demonstrate that GBSG-RD consistently outperforms several baseline algorithms. Finally, we extend the GBSGFRS model into a more flexible VGBSGFRS model by incorporating a variable precision parameter. Leveraging the approximation operators inherent in VGBSGFRS, we propose a granular form anomaly scoring metric for data instances and accordingly design a novel unsupervised anomaly detection algorithm, VGBSG-AD. Experimental evaluations on 16 diverse datasets show that VGBSG-AD achieves significantly higher detection accuracy compared to established benchmark methods.
Interval branch-and-bound (IBB) methods provide guaranteed enclosures of unknown but bounded domains for global optimization and parameter estimation problems, but their adoption is often limited by computational cost. This article examines how interval algorithms can be mapped to high-performance hardware and evaluates several parallelization strategies with respect to their speed and energy efficiency. Benchmark results indicate that vectorized GPU methods surpass sequential and brute-force approaches for higher-dimensional problems, whereas CPU parallelization remains competitive for small-scale tasks. We demonstrate the effectiveness of the vectorized approach using the example of battery cell parameter estimation and obtain narrow parameter enclosures within feasible execution times.
The Möbius transform has facilitated great progress in both theory and practical applications of fuzzy measures. As well as simplifying a number of calculations used for interpretation, the Möbius representation of certain families like k-additive fuzzy measures significantly reduces the number of defining parameters. For some other classes, like k-interactive fuzzy measures, we can use the less well-known co-Möbius representation, however this is not as intuitive or straightforward when it comes to interpreting its values. In this contribution we propose a modification to this calculation that we call the complement co-Möbius representation, which leads to more natural expressions, useful simplifications and insights. We provide the conversion formulas between each of the representations as well as some examples, highlighting k-interactive and plausibility measures in particular.
The paper presents new constructions of moderate deviation functions based on automorphisms and negation functions defined as in the symmetric interval [−1,1]. These constructions are then extended to a general closed interval [a,b]⊂R. Consequently, all results formulated for [−1,1] can be transferred to arbitrary intervals, yielding symmetric, idempotent aggregation functions. The paper also introduces moderate deviation functions that are non-continuous or not strictly monotonic, giving rise to quantile-type operators. An illustrative example demonstrates how aggregation functions derived from these moderate deviation functions can replace standard methods in the fusion of features extracted by Convolutional Neural Networks. Experiments indicate that several of these constructions are effective alternatives that, when optimised by gradient descent, can enhance model performance without significant computational overhead.
Outlier Detection (OD) has attracted wide attention due to its important applications in financial risk control, cybersecurity, and medical diagnosis. Clustering-based outlier detection identifies outliers by analyzing the relationship between data and clustering structures, offering strong intuitive and interpretable results. However, the performance of such methods heavily depends on clustering results, and they often fail to adequately account for the interference of outliers during clustering, thereby degrading detection performance. Moreover, these methods typically struggle to simultaneously detect different types of outliers, leading to missed outliers. Multi-kernel learning adaptively integrates multiple kernel functions to map data into an appropriate feature space, thereby alleviating the influence of outliers on cluster structures, improving clustering quality, and enhancing outlier detection performance. Based on these observations, this paper proposes a multi-kernel clustering-based outlier detection method. Specifically, a kernel representation composed of multiple kernel functions is first constructed, and the weights of each kernel are adaptively learned during the clustering process to obtain the clustering partition of samples. Then, based on the clustering results, the clusters are divided into large and small clusters. By jointly considering cluster size and sample deviation, an outlier score is defined to enable the simultaneous detection of different types of outlier samples. Experimental comparisons with state-of-the-art methods show that the proposed method achieves competitive AUC performance. The code is publicly available online at https://github.com/Tian468/MKCOD.
Reliable state estimation for autonomous systems in structured environments frequently depends on acoustic and ultrasonic range sensors. Unlike high-resolution LiDAR, these sensors possess a significant angular aperture, where the measurement follows a first echo principle returning the minimum distance to any obstacle within a sensing cone. This measurement model introduces non-differentiable ”min” operators that are inherently difficult to handle with classical probabilistic filters like the Extended Kalman Filter (EKF). This paper proposes a guaranteed state estimation framework based on the remoteness constraint, originally introduced as an inclusion test but limited by computational inefficiency. We present an implementation of the Remoteness Separator within the framework of separator algebra, enabling the use of efficient contraction techniques to prune the state space. A core contribution of this work is the derivation of an efficient implementation which does not generate fake boundaries, thus maintaining the topological integrity of the solution set and enhancing computational performance. The approach is evaluated through a state estimation benchmark in a structured environment, providing a mathematically guaranteed enclosure of the robot’s pose. This implementation is restricted to the two-dimensional case with obstacle segments, but the underlying principles could be extended to more complex geometries and higher dimensions.
Group Role Assignment (GRA) is a fundamental decision problem in Role-Based Collaboration (RBC), where agents are assigned to predefined roles to improve team performance. In interaction-intensive tasks, effective role assignment depends not only on individual–role qualification matching but also on cooperation–conflict relationships among team members. Existing GRA extensions still face two challenges: First, interaction information is typically derived from subjective, experience-dependent sources, yielding sparse relationship representations with limited transferability. Second, cooperation and conflict are commonly modeled through linear-additive formulations, which fail to adequately capture their asymmetric effects on actual team performance. To address these issues, this paper proposes an Interaction-Modulated Group Role Assignment (IM-GRA) model. It first infers latent cooperation–conflict tendencies from role-contextualized member features, then regulates the derived similarity information through a reconstruction mechanism grounded in three-way decision theory. Drawing upon Herzberg’s Two-Factor Theory, we construct a fractional nonlinear utility function that models cooperation as a motivating factor and conflict as a hygiene factor. The resulting nonlinear problem is then exactly reformulated as a mixed-integer linear program via variable substitution and McCormick-envelope-based linearization. Experiments on a real-world scenario and simulated instances show that IM-GRA achieves higher realized team performance than benchmark models under different scales and interaction densities.
Imbalanced data presents significant challenges in many real-world applications, where the minority class often holds critical importance but is underrepresented. It becomes more complex when the data involves multiple classes. Numerous methods are currently available for multi-class imbalance problems. However, most of them convert the problem into multiple binary classification tasks, making it challenging to fully capture the original distribution of multi-class data. This paper aims to address the problem by proposing a fuzzy rough set-based method, which focuses on the intrinsic relationships between the classes and features (class-feature relationships). To overcome the limitations of euclidean distance in imbalanced datasets which is sensitive to the scale of individual features, we introduce the standardized euclidean distance and propose a corresponding fuzzy rough set based on it. Using the lower and upper approximation operators, we define the intra-class and inter-class determinacies of samples for each individual feature, and fuse them into a predictive membership by weighting each feature’s importance. Combining this with a cost-sensitive strategy, we formulate a cost-weighted objective function that is trivially solvable via the classical gradient method, enabling efficient weight adaptation and the development of learning algorithms. After that, the prediction algorithm is proposed by using the optimal weights and the corresponding cost vector. Numerical experiments on 13 imbalanced datasets against 8 representative algorithms demonstrate the effectiveness of the proposed method.
In this paper, we address fundamental computational challenges within the framework of numerical analogies. In particular, we consider the computation of analogical powers, which somewhat quantify the analogical relationship between the four numbers. Prior work established the feasibility of using a bisection algorithm to search for this value, but the efficiency of this search is entirely dependent on having tight bounds.The latter motivates this paper’s first objective, namely, to precisely define the necessary search space. For that, we prove lower and upper bounds for the analogical power of a given numerical analogy that reliably minimize the search space for the bisection algorithm that ensures the exact computation of analogical powers with guaranteed termination. To avoid computation burdens, we also propose a closed-form approximation of analogical powers based on a generalized sigmoid function. Both the bounds and the approximation methods are rigorously assessed in terms of their conformance, deviation, and computational time complexity.The practical utility of our results is demonstrated through an application to image processing. We integrate both the exact (bound-enabled) and approximate computational methods into the process of “analogical pooling” for image reduction and reconstruction tasks, using the MNIST dataset as a benchmark. The experimental results provide a comparative analysis of the trade-offs between computation time and resulting image quality.This work significantly advances the feasibility and applicability of analogical reasoning in real-world applications and complex high-dimensional downstream tasks. It provides theoretical results immediately usable in these real-world applications.
This paper introduces a new wrapper called a gutter, designed to enclose sections of the solution set defined by underdetermined nonlinear equations. We demonstrate that, compared to existing wrappers, gutters can provide a more accurate outer approximation with lower computational cost. Several examples from the literature illustrate the efficiency and accuracy of this approach.
