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.