On the Continuity of Rotation Representations in Neural Networks

TL;DR

This paper explores the mathematical continuity of rotation representations in neural networks, revealing that common 3D rotation representations are discontinuous and proposing higher-dimensional continuous alternatives that improve learning performance.

Contribution

It introduces a formal definition of continuous rotation representations, demonstrates their existence in higher dimensions, and empirically shows their advantages in neural network applications.

Findings

Common 3D rotation representations are discontinuous in low dimensions.

Higher-dimensional continuous representations improve neural network training.

Empirical results show better performance of continuous representations in graphics and vision tasks.

Abstract

In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous representation, which can be helpful for training deep neural networks. We relate this to topological concepts such as homeomorphism and embedding. We then investigate what are continuous and discontinuous representations for 2D, 3D, and n-dimensional rotations. We demonstrate that for 3D rotations, all representations are discontinuous in the real Euclidean spaces of four or fewer dimensions. Thus, widely used representations such as quaternions and Euler angles are discontinuous and difficult for neural networks to learn. We show that the 3D rotations have continuous representations in 5D and 6D, which are more suitable for learning. We also present continuous representations for the general case of the n-dimensional rotation group SO(n). While our main focus is on rotations, we also show that our constructions apply to other groups such as the orthogonal group and similarity transforms. We finally present empirical results, which show that our continuous rotation representations outperform discontinuous ones for several practical problems in graphics and vision, including a simple autoencoder sanity test, a rotation estimator for 3D point clouds, and an inverse kinematics solver for 3D human poses.