  There is growing interest in using the close connection between differential
geometry and statistics to model smooth manifold-valued data. In particular,
much work has been done recently to generalize principal component analysis
(PCA), the method of dimension reduction in linear spaces, to Riemannian
manifolds. One such generalization is known as principal geodesic analysis
(PGA). This paper, in a novel fashion, obtains Taylor expansions in scaling
parameters introduced in the domain of objective functions in PGA. It is shown
this technique not only leads to better closed-form approximations of PGA but
also reveals the effects that scale, curvature and the distribution of data
have on solutions to PGA and on their differences to first-order tangent space
approximations. This approach should be able to be applied not only to PGA but
also to other generalizations of PCA and more generally to other intrinsic
statistics on Riemannian manifolds.
