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.