On Procrustes Analysis in Hyperbolic Space

TL;DR

This paper extends Procrustes analysis to hyperbolic space, providing a closed-form solution for noise-free cases and analyzing performance under measurement noise, advancing geometric alignment methods.

Contribution

It formulates the Procrustes problem in hyperbolic space and derives a closed-form solution, which was not previously available.

Findings

Closed-form solution for hyperbolic Procrustes problem

Analysis of method performance under measurement noise

Review of hyperbolic point set centering

Abstract

Congruent Procrustes analysis aims to find the best matching between two point sets through rotation, reflection and translation. We formulate the Procrustes problem for hyperbolic spaces, review the canonical definition of the center of point sets, and give a closed form solution for the optimal isometry for noise-free measurements. We also analyze the performance of the proposed method under measurement noise.