  Principal Component Analysis (PCA) is a workhorse of modern data science.
While PCA assumes the data conforms to Euclidean geometry, for specific data
types, such as hierarchical and cyclic data structures, other spaces are more
appropriate. We study PCA in space forms; that is, those with constant
curvatures. At a point on a Riemannian manifold, we can define a Riemannian
affine subspace based on a set of tangent vectors. Finding the optimal
low-dimensional affine subspace for given points in a space form amounts to
dimensionality reduction. Our Space Form PCA (SFPCA) seeks the affine subspace
that best represents a set of manifold-valued points with the minimum
projection cost. We propose proper cost functions that enjoy two properties:
(1) their optimal affine subspace is the solution to an eigenequation, and (2)
optimal affine subspaces of different dimensions form a nested set. These
properties provide advances over existing methods, which are mostly iterative
algorithms with slow convergence and weaker theoretical guarantees. We evaluate
the proposed SFPCA on real and simulated data in spherical and hyperbolic
spaces. We show that it outperforms alternative methods in estimating true
subspaces (in simulated data) with respect to convergence speed or accuracy,
often both.
