  In this paper, we provide some careful analysis of certain pathological
behavior of Euler angles and unit quaternions encountered in previous works
related to rotation representation in neural networks. In particular, we show
that for certain problems, these two representations will provably produce
completely wrong results for some inputs, and that this behavior is inherent in
the topological property of the problem itself and is not caused by unsuitable
network architectures or training procedures. We further show that previously
proposed embeddings of $\mathrm{SO}(3)$ into higher dimensional Euclidean
spaces aimed at fixing this behavior are not universally effective, due to
possible symmetry in the input causing changes to the topology of the input
space. We propose an ensemble trick as an alternative solution.
