Outlier detection refers to the identification of anomalous samples that deviate significantly from the distribution of normal data and has been extensively studied and used in a variety of practical tasks. However, most unsupervised outlier detection methods are carefully designed to detect specified outliers, while real-world data may be entangled with different types of outliers. In this study, we propose a fuzzy rough sets-based multi-scale outlier detection method to identify various types of outliers. Specifically, a novel fuzzy rough sets-based method that integrates relative fuzzy granule density is first introduced to improve the capability of detecting local outliers. Then, a multi-scale view generation method based on granular-ball computing is proposed to collaboratively identify group outliers at different levels of granularity. Moreover, reliable outliers and inliers determined by the three-way decision are used to train a weighted support vector machine to further improve the performance of outlier detection. The proposed method innovatively transforms unsupervised outlier detection into a semi-supervised classification problem and for the first time explores the fuzzy rough sets-based outlier detection from the perspective of multi-scale granular balls, allowing for high adaptability to different types of outliers. Extensive experiments carried out on both artificial and UCI datasets demonstrate that the proposed outlier detection method significantly outperforms the state-of-the-art methods, improving the results by at least 8.48% in terms of the Area Under the ROC Curve (AUROC) index. The source codes are released at https://github.com/Xiaofeng-Tan/MGBOD

Available subspace clustering methods often contain two stages, finding low-dimensional subspaces of data and then conducting clustering in the subspaces. Therefore, how to find the subspaces that better represent the original data becomes a research challenge. However, most of the reported methods are based on the premise that the contributions of different features are equal, which may not be ideal for real scenarios, i.e., the contributions of the important features may be overwhelmed by a large amount of redundant features. In this study, a weighted subspace fuzzy clustering (WSFC) model with a locality preservation mechanism is presented, which can adaptively capture the importance of different features, achieve an optimal lower-dimensional subspace, and perform fuzzy clustering simultaneously. Since each feature can be well quantified in terms of its importance, the proposed model exhibits the sparsity and robustness of fuzzy clustering. The intrinsic geometrical structures of data can also be preserved while enhancing the interpretability of clustering tasks. Extensive experimental results show that WSFC can allocate appropriate weights to different features according to data distributions and clustering tasks and achieve superior performance compared to other clustering models on real-world datasets.

The performance of k-means clustering is often degenerate when dealing with high-dimensional and noisy scenarios. In this study, an end-to-end robust clustering method with low-rank linear embedding techniques (RCLR) is presented in conjunction with k-means. Sparse coefficients and a space projection matrix can be simultaneously learned. The global structures and local neighborhood properties are well captured in the learning procedures. Both the processes of clustering and dimensionality reduction are realized at the same time. The notions of clustering, dimensionality reduction, low-rank representation, and local property preservation are seamlessly integrated into a unified model. The limitation of error accumulation encountered in the previous two-stage clustering framework involving low-rank representation can be alleviated. This is the first attempt to introduce both the global and local geometrical structures into k-means directly, as well L2,1-norm is used as a basic metric instead of the conventional F-norm to further improve the robustness and interpretation of the model. The superiority of the proposed RCLR method is demonstrated by extensive experiments completed on various well-known benchmark datasets.

Shadowed sets provide a meaningful description of information granules by abstracting the corresponding fuzzy sets into three categories: full acceptance, full rejection, and uncertain (represented by shadows). One of the main motivating points to derive shadowed sets from fuzzy sets is the determination and explanation of the separation thresholds based on a specific optimization mechanism. The available optimization objective functions are mainly discussed on semantic interpretations and their mathematical properties; constructive algorithms for optimal solutions have rarely been reported. In this paper, the continuous and convex properties of Pedrycz's optimization objective function to construct shadowed sets, as well as the existence and uniqueness of solution points, are analyzed in detail. It is demonstrated that different approximation region partitions would be generated even under the same optimization model, which requires further criteria to make the constructed shadowed sets well-defined. To address this limitation, the notions of passive and active constrained shadowed sets are introduced. A fast algorithm to obtain the proposed constrained shadowed sets is also designed based on the analyzed mathematical properties. Its performance is then illustrated by some typical fuzzy sets and some real data from the UCI repository.