What question did this study set out to answer?

The aim is to introduce and analyze a novel principal component analysis method in $$\Phi _\alpha$$-linear spaces.

May 26, 2026Open Access

Principal component analysis in _-linear spaces

Key Points

The aim is to introduce and analyze a novel principal component analysis method in $$\Phi _\alpha$$-linear spaces.
Introduces $$\Phi _\alpha$$-PCA as an isomorphic deformation of standard PCA with a hyperparameter $$\alpha$$.
Demonstrates the existence of U-statistics in $$\Phi _\alpha$$-linear spaces, similar to standard PCA statistics.
Conducts simulations and image applications to illustrate the advantages of $$\Phi _\alpha$$-PCA in noisy environments.
Establishes that the standard PCA is a special case of $$\Phi _\alpha$$-PCA when $$\alpha = 1$$.
Shows that $$\Phi _\alpha$$-PCA effectively handles noise and outliers in data analysis.
Simulations indicate improved performance of $$\Phi _\alpha$$-PCA over standard PCA in various practical applications.

Abstract

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

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