Fast Extended Inductive Robust Principal Component Analysis With Optimal Mean

Abstract

Inspired by the mean calculation of RPCA_OM and inductiveness of IRPCA, we first propose an inductive robust principal component analysis method with removing the optimal mean automatically, which is shorted as IRPCA_OM. Furthermore, IRPCA_OM is extended to Schatten- <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula> norm and a more general framework (i.e., EIRPCA_OM) is presented. The objective function of EIRPCA_OM includes two terms, the first term is a robust reconstruction error term constrained by an <inline-formula><tex-math notation="LaTeX">$\ell _{2,1}$</tex-math></inline-formula> -norm and the second term is a regularization term constrained by a Schatten- <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula> norm. The proposed EIRPCA_OM method is robust, inductive and accurate. However, on the high-dimensional data, it would spend a large computation cost in training stage. To this end, a fast version of EIRPCA_OM called as FEIRPCA_OM is proposed, and its basic idea is to eliminate the zero eigenvalues of data matrix. More importantly, an effective theoretical proof is presented to ensure that FEIRPCA_OM has faster processing speed than EIRPCA_OM when processing high-dimensional data, but without any performance loss. Based on it, we also can exchange the less performance loss for the higher computation efficiency by removing the small eigenvalues of data matrix. Experimental results on the public datasets demonstrate that FEIRPCA_OM works efficiently on the high-dimensional data.