R ₁-PCA: Rotational invariant L ₁-norm principal component analysis for robust subspace factorization

Principal component analysis (PCA) minimizes the sum of squared errors (L ₂-norm) and is sensitive to the presence of outliers. We propose a rotational invariant L ₁-norm PCA (R ₁-PCA). R ₁-PCA is similar to PCA in that (1) it has a unique global solution, (2) the solution are principal eigenvectors of a robust covariance matrix (re-weighted to soften the effects of outliers), (3) the solution is rotational invariant. These properties are not shared by the Li-norm PCA. A new subspace iteration algorithm is given to compute R ₁-PCA efficiently. Experiments on several real-life datasets show R ₁-PCA can effectively handle outliers. We extend R ₁-norm to K-means clustering and show that L ₁-norm K-means leads to poor results while R ₁-K-means outperforms standard K-means.