  This paper defines fair principal component analysis (PCA) as minimizing the
maximum mean discrepancy (MMD) between dimensionality-reduced conditional
distributions of different protected classes. The incorporation of MMD
naturally leads to an exact and tractable mathematical formulation of fairness
with good statistical properties. We formulate the problem of fair PCA subject
to MMD constraints as a non-convex optimization over the Stiefel manifold and
solve it using the Riemannian Exact Penalty Method with Smoothing (REPMS; Liu
and Boumal, 2019). Importantly, we provide local optimality guarantees and
explicitly show the theoretical effect of each hyperparameter in practical
settings, extending previous results. Experimental comparisons based on
synthetic and UCI datasets show that our approach outperforms prior work in
explained variance, fairness, and runtime.
