Optimization over Geodesics for Exact Principal Geodesic Analysis

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.