  We present a 3D capsule module for processing point clouds that is
equivariant to 3D rotations and translations, as well as invariant to
permutations of the input points. The operator receives a sparse set of local
reference frames, computed from an input point cloud and establishes end-to-end
transformation equivariance through a novel dynamic routing procedure on
quaternions. Further, we theoretically connect dynamic routing between capsules
to the well-known Weiszfeld algorithm, a scheme for solving \emph{iterative
re-weighted least squares} (IRLS) problems with provable convergence
properties. It is shown that such group dynamic routing can be interpreted as
robust IRLS rotation averaging on capsule votes, where information is routed
based on the final inlier scores. Based on our operator, we build a capsule
network that disentangles geometry from pose, paving the way for more
informative descriptors and a structured latent space. Our architecture allows
joint object classification and orientation estimation without explicit
supervision of rotations. We validate our algorithm empirically on common
benchmark datasets.
