On Procrustes Analysis in Hyperbolic Space

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 mass for a point set, and give a closed-form solution for the optimal isometry between noise-free point sets. Our algorithm is analogous to the Euclidean Procrustes analysis, with centering and rotation replaced by their hyperbolic counterparts. When the data is corrupted with noise, our algorithm computes a sub-optimal alignment. We thus propose a gradient-based fine-tuning method to improve the matching accuracy.

Introduction

In GREEK mythology, Procrustes was a robber who lived in Attica and deformed his victims to match the size of his bed. In 1962, Hurley and Catell used the story of Procrustes to describe a point set matching problem in Euclidean spaces [1], stated below.