Optimization over Geodesics for Exact Principal Geodesic Analysis

TL;DR

This paper introduces numerical methods for optimization over geodesics in non-linear spaces, enabling exact Principal Geodesic Analysis and providing insights into the geometry of data manifolds.

Contribution

It develops a novel numerical approach for optimization over geodesics, facilitating exact PGA computation and geometric analysis in non-linear spaces.

Findings

Exact PGA differs from linearized methods on synthetic data.

Numerical integration of Jacobi fields estimates sectional curvatures.

Upper bounds for injectivity radii are derived.

Abstract

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.