Robust local K-proximal plane clustering based on L2,1-norm minimization

K-proximal plane clustering (kPPC) cluster data points to the center points and local k-proximal plane clustering (LkPPC) uses the combination of hyperplane and points as the cluster center to localize the hyperplane. However, the l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{2}$$\end{document}-norm is employed to group the data into corresponding clusters, which is sensitive to outliers because of the square operation. Many previous works chose to use l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{1}$$\end{document}-norm instead of l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{2}$$\end{document}-norm to improve robustness. However, this approach has limited improvement in the robustness of outlier, and the solution of l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{1}$$\end{document}-norm mostly uses a greedy algorithm search strategy, which is easy to fall into local optimization and consumes a long time. In this paper, we propose a clustering method using l2,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{2,1}$$\end{document}-norm, named RLkPPC. To solve the objective function, we combine an efficient iterative optimization algorithm with the Lagrange multiplier method, and on this basis, propose a non-greedy weighted iterative optimization algorithm for solving the l2,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{2,1}$$\end{document}-norm minimum problem. Compared to existing methods, the advantage of our method is: (1) similar to LkPPC, it has a clear geometric explanation; (2) it has good robustness and a stronger ability to resist the influence of outliers; (3) it uses a non-greedy weighted iterative optimization algorithm, prevent falling into local optima. The experimental results on some artificial and benchmark datasets indicate that our algorithm has the robustness and clustering accuracy advantages.