Abstract

us En li Manifold-valued datasets are widely encountered in gimany computer vision tasks. A non-linear analog of trthe PCA algorithm, called the Principal Geodesic Analubysis (PGA) algorithm suited for data lying on Riemantonian manifolds was reported in literature a decade ago. LeSince the objective function in the PGA algorithm is cihighly non-linear and hard to solve efficiently in genmaeral, researchers have proposed a linear approximation. Though this linear approximation is easy to compute, it lilacks accuracy especially when the data exhibits a large povariance. Recently, an alternative called the exact PGA liwas proposed which tries to solve the optimization withpuout any linearization. For general Riemannian manilafolds, though it yields a better accuracy than the origfuinal (linearized) PGA, for data that exhibit large varithance, the optimization is not computationally efficient. anIn this paper, we propose an efficient exact PGA altogorithm for constant curvature Riemannian manifolds it(CCM-EPGA). The CCM-EPGA algorithm differs sigisnificantly from existing PGA algorithms in two aspects, op(i) the distance between a given manifold-valued data fopoint and the principal submanifold is computed anmualytically and thus no optimization is required as in CCthe existing methods. (ii) Unlike the existing PGA alofgorithms, the descent into codimension-1 submanifolds stdoes not require any optimization but is accomplished pethrough the use of the Rimeannian inverse Exponential thmap and the parallel transport operations. We present (Stheoretical and experimental results for constant curtivature Riemannian manifolds depicting favorable perreformance of the CCM-EPGA algorithm compared to apexisting PGA algorithms. We also present data reconsistruction from the principal components which has not Imbeen reported in literature in this setting. Vi 13976