Principal component analysis in Φ _α-linear spaces

This paper introduces a principal component analysis (Φ _α-PCA) in a topological structure called Φ _α-linear spaces. The Φ _α-PCA consists in an isomorphic deformation of the usual PCA through a hyperparameter α. It is shown that the statistics employed in standard PCA (cosines and correlations) exist in Φ _α-linear spaces, these are U-statistics. As the regular PCA is a special case of Φ _α-PCA when α = 1, simulations and applications on images are provided to outline the relevance of the Φ _α-PCA in various settings and specifically in the presence of noise and outliers.