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Existing hierarchical clustering (HC) algorithms generally depend on the Euclidean characteristic metric (Euclidean distance, Manhattan distance, Chebyshev distance, etc.) on Euclidean space to describe the similarity between objects, which makes the clustering process oriented to data sets with uniform and regular distribution in Euclidean space. Although such methods can visually distinguish the cluster distribution of data, it is not effective for the data sets which are densely distributed, interlaced and complex in Euclidean space. As a scalable, efficient and robust method, granular computing generally analyzes data from the perspective of similarity and proximity. In consideration of the advantages of granular computing in extracting data information from a multi-level perspective, in order to reduce the limitations of HC methods based on Euclidean features on non-Euclidean data, this paper proposes a novel HC method based on non-Euclidean feature structure. First, this paper constructs the similarity between objects based on K -dominance granularity and neighborhood search, and considers the environmental information of data points from both global and local perspectives. Secondly, a new HC method based on non-Euclidean feature structure is designed on the basis of the similarity measurement constructed in this paper. Finally, through comparative analysis, the experimental results prove that our method can more accurately identify the densely distributed and interlaced data sets in Euclidean space; it is significantly better than comparison algorithms using different Euclidean features to measure similarity; it has good robustness when additional Gaussian noise is added. [ABSTRACT FROM AUTHOR]

Interval reciprocal preference relations (IRPRs) have been extensively applied to real-life decision-making problems. This paper aims at introducing two new decision-making methods with IRPRs. To overcome drawbacks of several extant order relations of intervals, a new admissible order relation of intervals is proposed. An algorithm is then introduced to rank a series of intervals. We define a new consistency index and the satisfactory consistency of IRPRs. For improving the consistency level of IRPRs, a linear programming model is built. Subsequently, a new individual decision-making (IDM) method with an IRPR is introduced. For group decision making, a group consensus index is proposed. To improve the group consensus degree, an interactive convergent iterative algorithm is designed. Decision makers' weights are determined by combining the logarithmic Manhattan distance between two IRPRs with the consistency indices of IRPRs. Accordingly, a novel consistent and consensus-based group decision-making (GDM) method with IRPRs is presented. Eventually, a decision support system is developed based on the proposed IDM method and GDM method. Illustrative examples and simulation experiments are provided to illustrate the superiority of the proposed IDM method and GDM method. [ABSTRACT FROM AUTHOR]

Existing robust non-negative matrix factorization methods fail to achieve data recovery and learn a robust representation. This is because these methods suppose that outliers and noise of the original data are the Gaussian distribution. In this paper, we propose a robust non-negative matrix model, called robust Manhattan non-negative matrix factorization, which can handle various noise (e.g. Gaussian noise, Salt and Pepper noise or Contiguous Occlusion). Different from previous robust non-negative matrix factorization models, we utilize mean filter and matrix completion as additional constraints to recover the corrupted data from normal data or neighbouring corrupted data, and achieve a robust low-dimensional representation by Manhattan non-negative matrix factorization. We theoretically compare the robustness of our proposed model with other non-negative matrix factorization models and theoretically prove the effectiveness of the proposed algorithm. Extensive experimental results on the image dataset containing noise and outliers validate the robustness and effectiveness of our proposed model for image recovery and representation. [ABSTRACT FROM AUTHOR]