Scale and curvature effects in principal geodesic analysis

TL;DR

This paper develops Taylor expansion techniques to analyze how scale and curvature influence principal geodesic analysis on Riemannian manifolds, improving approximation accuracy and understanding of intrinsic statistical methods.

Contribution

It introduces a novel Taylor expansion approach for PGA, revealing the effects of scale, curvature, and data distribution on solutions and their tangent space approximations.

Findings

Enhanced closed-form approximations of PGA.

Quantitative insights into scale and curvature effects.

Applicability to other intrinsic Riemannian statistics.

Abstract

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.