Manifold-valued datasets are widely encountered in many computer vision
tasks. A non-linear analog of the PCA, called the Principal Geodesic Analysis
(PGA) suited for data lying on Riemannian manifolds was reported in literature
a decade ago. Since the objective function in PGA is highly non-linear and hard
to solve efficiently in general, researchers have proposed a linear
approximation. Though this linear approximation is easy to compute, it lacks
accuracy especially when the data exhibits a large variance. Recently, an
alternative called exact PGA was proposed which tries to solve the optimization
without any linearization. For general Riemannian manifolds, though it gives
better accuracy than the original (linearized) PGA, for data that exhibit large
variance, the optimization is not computationally efficient. In this paper, we
propose an efficient exact PGA for constant curvature Riemannian manifolds
(CCM-EPGA). CCM-EPGA differs significantly from existing PGA algorithms in two
aspects, (i) the distance between a given manifold-valued data point and the
principal submanifold is computed analytically and thus no optimization is
required as in existing methods. (ii) Unlike the existing PGA algorithms, the
descent into codimension-1 submanifolds does not require any optimization but
is accomplished through the use of the Rimeannian inverse Exponential map and
the parallel transport operations. We present theoretical and experimental
results for constant curvature Riemannian manifolds depicting favorable
performance of CCM-EPGA compared to existing PGA algorithms. We also present
data reconstruction from principal components and directions which has not been
presented in literature in this setting.

本文提出了一种适用于常曲率黎曼流形的精确 PGA 算法，它通过使用黎曼逆指数映射和平行传输操作，直接计算表征数据点和主子流形之间距离的分析量，并无需操作或优化，实现比现有 PGA 算法更高的计算效率和更好的精度，并展示了它在数据重建方面的成功应用。