T-product based ℓ1-norm tensor principal component analysis and a finite-step convergence algorithm

T-product based tensor principal component analysis (${\ell }_2$-tPCA) was used for dimensionality reduction, data preprocessing, compression, and visualization of multivariate data. However, ${\ell }_2$-tPCA may amplify the influence of outliers and large-magnitude noise. To explore robustness against heavily corrupted third-order data, we consider the ${\ell }_1$-norm tPCA model (${\ell }_1$-tPCA). We develop an effective proximal alternating maximization method and prove that within finitely many steps, the algorithm stops at a point satisfying certain optimality conditions. Numerical experiments on color face reconstruction and recognition demonstrate the efficiency of the proposed algorithms, confirming that ${\ell }_1$-tPCA is more resilient to outliers compared to ${\ell }_2$-tPCA.