Principal arc analysis on direct product manifolds

TL;DR

This paper introduces principal arc analysis, a novel method for capturing nonlinear modes of variation in high-dimensional data on direct product manifolds, improving visualization and understanding of complex data structures.

Contribution

It presents a new approach for low-dimensional representation of data on direct product manifolds, capturing nonlinear variation more effectively than previous manifold PCA extensions.

Findings

Captures variation along nongeodesic arcs on spheres in a single mode

Outperforms previous methods in representing nonlinear modes

Demonstrates effectiveness on medial representations in image analysis

Abstract

We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.