  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.
