Due to its strong performance in handling uncertain and ambiguous data, the fuzzy k-nearest-neighbor method (FKNN) has realized substantial success in a wide variety of applications. However, its classification performance would be heavily deteriorated if the number k of nearest neighbors was unsuitably fixed for each testing sample. This study examines the feasibility of using only one fixed k value for FKNN on each testing sample. A novel FKNN-based classification method, namely, fuzzy KNN method with adaptive nearest neighbors (A-FKNN), is devised for learning a distinct optimal k value for each testing sample. In the training stage, after applying a sparse representation method on all training samples for reconstruction, A-FKNN learns the optimal k value for each training sample and builds a decision tree (namely, A-FKNN tree) from all training samples with new labels (the learned optimal k values instead of the original labels), in which each leaf node stores the corresponding optimal k value. In the testing stage, A-FKNN identifies the optimal k value for each testing sample by searching the A-FKNN tree and runs FKNN with the optimal k value for each testing sample. Moreover, a fast version of A-FKNN, namely, FA-FKNN, is designed by building the FA-FKNN decision tree, which stores the optimal k value with only a subset of training samples in each leaf node. Experimental results on 32 UCI datasets demonstrate that both A-FKNN and FA-FKNN outperform the compared methods in terms of classification accuracy, and FA-FKNN has a shorter running time.

As an exemplar-based clustering method, the well-known density peaks clustering (DPC) heavily depends on the computation of kernel-based density peaks, which incurs two issues: (1) whether kernel-based density can well facilitate a large variety of data, including cases where ambiguity and uncertainty of the assignment of the data points to their clusters may exist, and (2) whether the concept of density peaks can be interpreted and manipulated from the perspective of soft partitions (e.g., fuzzy partitions) to achieve enhanced clustering performance. In this study, in order to provide flexible adaptability for tackling ambiguity and uncertainty in clustering, a new concept of fuzzy peaks is proposed to express the density of a data point as the fuzzy-operator-based coupling of the fuzzy distances between a data point and its neighbors. As a fuzzy variant of DPC, a novel fuzzy density peaks clustering method FDPC based on fuzzy operators (especially S-norm operators) is accordingly devised along with the same algorithmic framework of DPC. With an appropriate choice of a fuzzy operator with its associated tunable parameter for a clustering task, FDPC can indeed inherit the advantage of fuzzy partitions and simultaneously provide flexibility in enhancing clustering performance. The experimental results on both synthetic and real datasets demonstrate that the proposed method outperforms or at least remains comparable to the comparative methods in clustering performance by choosing appropriate parameters in most cases.

Motivated by both the commonly used “from wholly coarse to locally fine” cognitive behavior and the recent finding that simple yet interpretable linear regression model should be a basic component of a classifier, a novel hybrid ensemble classifier called hybrid Takagi–Sugeno–Kang fuzzy classifier (H-TSK-FC) and its residual sketch learning (RSL) method are proposed. H-TSK-FC essentially shares the virtues of both deep and wide interpretable fuzzy classifiers and simultaneously has both feature-importance-based and linguistic-based interpretabilities. RSL method is featured as follows: 1) a global linear regression subclassifier on all original features of all training samples is generated quickly by the sparse representation-based linear regression subclassifier training procedure to identify/understand the importance of each feature and partition the output residuals of the incorrectly classified training samples into several residual sketches; 2) by using both the enhanced soft subspace clustering method (ESSC) for the linguistically interpretable antecedents of fuzzy rules and the least learning machine (LLM) for the consequents of fuzzy rules on residual sketches, several interpretable Takagi–Sugeno–Kang (TSK) fuzzy subclassifiers are stacked in parallel through residual sketches and accordingly generated to achieve local refinements; and 3) the final predictions are made to further enhance H-TSK-FC’s generalization capability and decide which interpretable prediction route should be used by taking the minimal-distance-based priority for all the constructed subclassifiers. In contrast to existing deep or wide interpretable TSK fuzzy classifiers, benefiting from the use of feature-importance-based interpretability, H-TSK-FC has been experimentally witnessed to have faster running speed and better linguistic interpretability (i.e., fewer rules and/or TSK fuzzy subclassifiers and smaller model complexities) yet keep at least comparable generalization capability.