Deep Regression on Manifolds: A 3D Rotation Case Study

TL;DR

This paper investigates differentiable mappings from Euclidean spaces to non-Euclidean manifolds, focusing on 3D rotations, and evaluates their properties and effectiveness for gradient-based learning tasks.

Contribution

It establishes desirable properties for such mappings, emphasizes the importance of pre-image connectivity, and compares various mappings for 3D rotation regression.

Findings

Procrustes orthonormalization-based mapping performs best.

Rotation vector representation is suitable for small angles.

Local linearity of mappings is crucial for performance.

Abstract

Many machine learning problems involve regressing variables on a non-Euclidean manifold -- e.g. a discrete probability distribution, or the 6D pose of an object. One way to tackle these problems through gradient-based learning is to use a differentiable function that maps arbitrary inputs of a Euclidean space onto the manifold. In this paper, we establish a set of desirable properties for such mapping, and in particular highlight the importance of pre-images connectivity/convexity. We illustrate these properties with a case study regarding 3D rotations. Through theoretical considerations and methodological experiments on a variety of tasks, we review various differentiable mappings on the 3D rotation space, and conjecture about the importance of their local linearity. We show that a mapping based on Procrustes orthonormalization generally performs best among the mappings considered, but that a rotation vector representation might also be suitable when restricted to small angles.