  Fair principal component analysis (FPCA), a ubiquitous dimensionality
reduction technique in signal processing and machine learning, aims to find a
low-dimensional representation for a high-dimensional dataset in view of
fairness. The FPCA problem involves optimizing a non-convex and non-smooth
function over the Stiefel manifold. The state-of-the-art methods for solving
the problem are subgradient methods and semidefinite relaxation-based methods.
However, these two types of methods have their obvious limitations and thus are
only suitable for efficiently solving the FPCA problem in special scenarios.
This paper aims at developing efficient algorithms for solving the FPCA problem
in general, especially large-scale, settings. In this paper, we first transform
FPCA into a smooth non-convex linear minimax optimization problem over the
Stiefel manifold. To solve the above general problem, we propose an efficient
alternating Riemannian/projected gradient descent ascent (ARPGDA) algorithm,
which performs a Riemannian gradient descent step and an ordinary projected
gradient ascent step at each iteration. We prove that ARPGDA can find an
$\varepsilon$-stationary point of the above problem within
$\mathcal{O}(\varepsilon^{-3})$ iterations. Simulation results show that,
compared with the state-of-the-art methods, our proposed ARPGDA algorithm can
achieve a better performance in terms of solution quality and speed for solving
the FPCA problems.
