A Novel Approach for Fair Principal Component Analysis Based on Eigendecomposition

Guilherme Dean Pelegrina, Leonardo Tomazeli Duarte

Abstract: Principal component analysis (PCA), a ubiquitous dimensionality reduction technique in signal processing, searches for a projection matrix that minimizes the mean squared error between the reduced dataset and the original one. As the classical PCA is not tailored to address concerns related to fairness, its application to actual problems may lead to disparity in the reconstruction errors of different groups (e.g., men and women, whites and blacks, etc.), with potentially harmful consequences. For instance, in terms of quality of representation in the projected space, one may retain more information from a specific group (e.g., men) instead of another one (e.g., women), which may introduce bias towards sensitive groups. Although several fair versions of PCA have been proposed recently, there still remains a fundamental gap in the search for algorithms that are simple enough to be deployed in real systems. Moreover, the considered fairness measure does not minimize, necessarily, the reconstruction errors of different groups. To address this, we propose a novel PCA algorithm which tackles fairness issues by means of a simple strategy comprising a 1-D search which exploits the closed-form solution of PCA. As attested by numerical experiments, the proposal can significantly improve fairness, by reducing disparities in reconstruction errors, with a very small loss in the overall reconstruction error and without resorting to complex optimization schemes. Moreover, our findings are consistent in several real situations as well as in scenarios with both unbalanced and balanced datasets.