Optimization over Geodesics for Exact Principal Geodesic Analysis

Understand

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.