Efficient Quasi-Geodesics on the Stiefel Manifold

TL;DR

This paper introduces new representations of quasi-geodesics on the Stiefel manifold, improving computational efficiency and accuracy for geodesic-related problems in data processing tasks.

Contribution

It derives a new representation for a known type of quasi-geodesics and proposes a novel kind of quasi-geodesics that better approximate Riemannian geodesics.

Findings

New large-scale computation-friendly representation of quasi-geodesics

A new quasi-geodesic type closer to true geodesics

Enhanced efficiency in geodesic endpoint problems

Abstract

Solving the so-called geodesic endpoint problem, i.e., finding a geodesic that connects two given points on a manifold, is at the basis of virtually all data processing operations, including averaging, clustering, interpolation and optimization. On the Stiefel manifold of orthonormal frames, this problem is computationally involved. A remedy is to use quasi-geodesics as a replacement for the Riemannian geodesics. Quasi-geodesics feature constant speed and covariant acceleration with constant (but possibly non-zero) norm. For a well-known type of quasi-geodesics, we derive a new representation that is suited for large-scale computations. Moreover, we introduce a new kind of quasi-geodesics that turns out to be much closer to the Riemannian geodesics.