Robust PCAs and PCA Using Generalized Mean

In this chapter, a robust principal component analysis (PCA) is described, which can overcome the problem that PCA is prone to outliers included in training set. Different from the other alternatives which commonly replace \(L_{2}\)-norm by other distance measures, our method alleviates the negative effect of outliers using the characteristic of the generalized mean keeping the use of the Euclidean distance. The optimization problem based on the generalized mean is solved by a novel method. We also present a generalized sample mean, which is a generalization of the sample mean, to estimate a robust mean in the presence of outliers. The proposed method shows better or equivalent performance than the conventional PCAs in various problems such as face reconstruction, clustering, and object categorization.