$\ell _{2,p}$ -Norm Based PCA for Image Recognition.

Recently, many ℓ1-norm-based PCA approaches have been developed to improve the robustness of PCA. However, most existing approaches solve the optimal projection matrix by maximizing ℓ1-norm-based variance and do not best minimize the reconstruction error, which is the true goal of PCA. Moreover, they do not have rotational invariance. To handle these problems, we propose a generalized robust metric learning for PCA, namely, ℓ2,p-PCA, which employs ℓ2,p -norm as the distance metric for reconstruction error. The proposed method not only is robust to outliers but also retains PCA's desirable properties. For example, the solutions are the principal eigenvectors of a robust covariance matrix and the low-dimensional representation have rotational invariance. These properties are not shared by ℓ1-norm-based PCA methods. A new iteration algorithm is presented to solve ℓ2,p-PCA efficiently. Experimental results illustrate that the proposed method is more effective and robust than PCA, PCA-L1 greedy, PCA-L1 nongreedy, and HQ-PCA.