Deep Regression on Manifolds: A 3D Rotation Case Study
This addresses a fundamental challenge in machine learning for tasks involving manifold-valued data, but it is incremental as it builds on existing mapping methods.
The paper tackled the problem of regressing variables on non-Euclidean manifolds, such as 3D rotations, by establishing desirable properties for differentiable mappings and showing that Procrustes orthonormalization generally performs best, with rotation vectors suitable for small angles.
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.