  In fields ranging from computer vision to signal processing and statistics,
increasing computational power allows a move from classical linear models to
models that incorporate non-linear phenomena. This shift has created interest
in computational aspects of differential geometry, and solving optimization
problems that incorporate non-linear geometry constitutes an important
computational task. In this paper, we develop methods for numerically solving
optimization problems over spaces of geodesics using numerical integration of
Jacobi fields and second order derivatives of geodesic families. As an
important application of this optimization strategy, we compute exact Principal
Geodesic Analysis (PGA), a non-linear version of the PCA dimensionality
reduction procedure. By applying the exact PGA algorithm to synthetic data, we
exemplify the differences between the linearized and exact algorithms caused by
the non-linear geometry. In addition, we use the numerically integrated Jacobi
fields to determine sectional curvatures and provide upper bounds for
injectivity radii.
