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