A Pure L1-norm Principal Component Analysis.

The norm has been applied in numerous variations of principal component analysis (PCA). -norm PCA is an attractive alternative to traditional -based PCA because it can impart robustness in the presence of outliers and is indicated for models where standard Gaussian assumptions about the noise may not apply. Of all the previously-proposed PCA schemes that recast PCA as an optimization problem involving the norm, none provide globally optimal solutions in polynomial time. This paper proposes an -norm PCA procedure based on the efficient calculation of the optimal solution of the -norm best-fit hyperplane problem. We present a procedure called -PCA* based on the application of this idea that fits data to subspaces of successively smaller dimension. The procedure is implemented and tested on a diverse problem suite. Our tests show that -PCA* is the indicated procedure in the presence of unbalanced outlier contamination.