CONIX Publication

In neural networks, it is often desirable to work with var- ious representations of the same space. For example, the 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continu- ous representation, which can be helpful for the learning of the network. We relate this to the definition of topo- logical equivalence. We then investigate what are contin- uous and discontinuous representations for 2D, 3D, and n- dimensional rotations. We demonstrate that for the 3D rota- tions, all representations are discontinuous in four or fewer dimension real Euclidean space. Thus, widely used repre- sentations such as quaternions and Euler angles are discon- tinuous. For the general case of the n dimensional rotation group SO(n), we present representations that are continu- ous and make the learning process easier. We use this result to show that the 3D rotations have continuous representa- tions in 5D and 6D. Although we focus mainly on rotations, we also show that our constructions apply to other groups such as the orthogonal group and the similarity transforms. We finally present empirical results that show that our con- tinuous rotation representations outperform discontinuous ones for several problems including a simple autoencoder sanity test, rotation estimation for 3D point clouds, and in- verse kinematics solver for 3D human poses.