Tensor Robust Principal Component Analysis via Tensor Fibered Rank and \({\boldsymbol{{l_p}}}\) Minimization
Kaixin Gao, Zheng‐Hai Huang
Abstract
.Tensor robust principal component analysis (TRPCA) is an important method to handle high-dimensional data and has been widely used in many areas. In this paper, we mainly focus on the TRPCA problem based on tensor fibered rank for sparse noise removal, which aims to recover the low-fibered-rank tensor from grossly corrupted observations. Usually, the \(l_1\) -norm is used as a convex approximation of tensor rank, but it is essentially biased and fails to achieve the best estimation performance. Therefore, we first propose a novel nonconvex model named \(\textrm{TRPCA}_p\) , in which the \(l_p\) norm ( \(0\lt p\lt 1\) ) is adopted to approximate tensor fibered rank and measure sparsity. Then, an error bound of the estimator of \(\textrm{TRPCA}_p\) is established and this error bound can be better than those of similar models based on Tucker rank or tubal rank. Further, we use the alternating direction method of multipliers to solve \(\textrm{TRPCA}_p\) and provide convergence guarantee. Finally, extensive experiments on color images, videos, and hyperspectral images demonstrate the effectiveness of the proposed method.Keywordstensor robust principal component analysisnonconvex approximationADMMtensor fibered rankMSC codes15A6990C26