Double Nonconvex Tensor Robust Kernel Principal Component Analysis and Its Visual Applications

Abstract

Tensor robust principal component analysis (TRPCA), as a popular linear low-rank method, has been widely applied to various visual tasks. The mathematical process of the low-rank prior is derived from the linear latent variable model. However, for nonlinear tensor data with rich information, their nonlinear structures may break through the assumption of low-rankness and lead to the large approximation error for TRPCA. Motivated by the latent low-dimensionality of nonlinear tensors, the general paradigm of the nonlinear tensor plus sparse tensor decomposition problem, called tensor robust kernel principal component analysis (TRKPCA), is first established in this paper. To efficiently tackle TRKPCA problem, two novel nonconvex regularizers the kernelized tensor Schatten- $p$ norm (KTSPN) and generalized nonconvex regularization are designed, where the former KTSPN with tighter theoretical support adequately captures nonlinear features (i.e., implicit low-rankness) and the latter ensures the sparser structural coding, guaranteeing more robust separation results. Then by integrating their strengths, we propose a double nonconvex TRKPCA (DNTRKPCA) method to achieve our expectation. Finally, we develop an efficient optimization framework via the alternating direction multiplier method (ADMM) to implement the proposed nonconvex kernel method. Experimental results on synthetic data and several real databases show the higher competitiveness of our method compared with other state-of-the-art regularization methods. The code has been released in our ResearchGate homepage: https://www.researchgate.net/publication/397181729_DNTRKPCA_code