Averaging on Manifolds by Embedding Algorithm

TL;DR

This paper introduces an embedding-based algorithm for finding critical points of cost functions on manifolds, simplifying computations by working in Euclidean space and demonstrating its application to SO(3) averaging problems.

Contribution

The paper presents a novel embedding algorithm that enables gradient computations in Euclidean space for manifold optimization problems, simplifying existing methods.

Findings

Algorithm successfully computes critical points on manifolds.

Simplifies calculations by using Cartesian coordinates.

Recovers known results in SO(3) averaging.

Abstract

We will propose a new algorithm for finding critical points of cost functions defined on a differential manifold. We will lift the initial cost function to a manifold that can be embedded in a Riemannian manifold (Euclidean space) and will construct a vector field defined on the ambient space whose restriction to the embedded manifold is the gradient vector field of the lifted cost function. The advantage of this method is that it allows us to do computations in Cartesian coordinates instead of using local coordinates and covariant derivatives on the initial manifold. We will exemplify the algorithm in the case of SO(3) averaging problems and will rediscover a few well known results that appear in literature.