Formulation of an unbiased version of principal component analysis based on a geometric approach

Dimensionality reduction is widely used in machine learning (ML), allowing complex datasets to be represented by means of a reduced number of attributes. However, classical methods such as Principal Component Analysis (PCA) present limitations regarding fairness in data representation, since projection may introduce distortions between samples from distinct groups. Although there are approaches that incorporate fairness criteria into PCA, these strategies focus on reconstruction quality, neglecting geometric aspects of the projected space. To address this gap, a multi-objective reformulation of PCA is proposed, based on a fairness criterion defined by the normalized difference between the areas of the convex hulls of two groups. The approach was evaluated in synthetic scenarios, and the results show a trade-off between reconstruction and fairness, such that the proposed multi-objective analysis enables the attainment of more balanced and interpretable visual representations.