A Unified Framework With Capped Tensor Norm Minimization for Multiview Subspace Learning

In recent years, multi-view subspace clustering (MVSC) has emerged as a powerful tool for high-dimensional data analysis, due to its ability to exploit intrinsic structures across multiple feature spaces. However, existing MVSC methods still suffer from two shortcomings: 1) the prevalent use of convex low-rank approximations inadequately exploits complementary subspace information, and 2) the neglect of intrinsic geometric relationships and the execution of spectral clustering as a separate step often lead to suboptimal solutions. To alleviate these problems, in this paper we propose a robust and effective model for the MVSC task by jointly incorporating the capped norm, which not only effectively captures the intrinsic global low-rank structure and suppresses the influence of feature outliers by utilizing capped tensor nuclear norm (CTNN), but also integrates the hyper- Laplacian regularization and the spectral embedding through joint optimization to organically benefit from each other. Then, a high computational efficiency algorithm is developed under the alternating direction method of multipliers (ADMM) framework to solve the proposed model. Meanwhile, rigorous mathematical analysis shows that the global optimum of the CTNN regularized least squares subproblem can be analytically determined in closed-form. More importantly, the convergence of the proposed algorithm is theoretically established by proving that the sequence it generates converges to a desirable Karush-Kuhn-Tucker (KKT) point. Finally, extensive experiments on various datasets demonstrate the superiority of the proposed model.