Principal Component Analysis Based on T${\ell }_1$-norm Maximization

Abstract

Classical principal component analysis (PCA) may suffer from the sensitivity to outliers and noise. Therefore PCA based on ${\ell }_1$-norm and ${\ell }_p$-norm ($0<p<1$) have been studied. Among them, the ones based on ${\ell }_p$-norm seem to be most interesting from the robustness point of view. However, their numerical performance is not satisfactory. Note that, although T${\ell }_1$-norm is similar to ${\ell }_p$-norm ($0<p<1$) in some sense, it has the stronger suppression effect to outliers and better continuity. So PCA based on T${\ell }_1$-norm is proposed in this paper. Our numerical experiments have shown that its performance is superior than PCA-${\ell }_p$ and ${\ell }_p$SPCA as well as PCA, PCA-${\ell }_1$ obviously.