Robust jointly sparse fast fuzzy clustering via ternary-tree-based anchor graph

Traditional partition-based fuzzy clustering algorithms are widely used for revealing possible hidden structures in data. However, high computational cost limits their applications in large-scale and high-dimensional data. Moreover, most fuzzy clustering algorithms are sensitive to noise. To tackle these issues, a robust jointly sparse fast fuzzy clustering algorithm via anchor graph (RSFCAG) is proposed and analyzed in this paper. Specifically, we first propose a fast k-means method integrated shadowed set and balanced ternary tree, which serves as a fast hierarchical clustering approach by partitioning every cluster into three subclusters at each layer (3KHK). 3KHK can quickly obtain the anchor set and its optimization is solved fast by the simplex method, which also captures the ambiguity and uncertainty between clusters in large-scale clustering tasks. Second, a similarity matrix learning approach based on possibilistic neighbors is further proposed to get a robust similarity graph, which strengthens the ability of fuzzy clustering to handle large-scale data. Furthermore, the orthogonal projection matrix is integrated into the RSFCAG framework to transform the original high-dimensional space into low-dimensional space. Finally, the $L_{2,1}$-norm loss and regularization are integrated into the joint algorithm RSFCAG, which is solved optimally by block coordinate technique, to enhance the robustness and interpretability of the fuzzy clustering process. The experimental results demonstrate the effectiveness and efficiency of our proposed method in most of benchmark datasets.