  3D Convolutional Neural Networks are sensitive to transformations applied to
their input. This is a problem because a voxelized version of a 3D object, and
its rotated clone, will look unrelated to each other after passing through to
the last layer of a network. Instead, an idealized model would preserve a
meaningful representation of the voxelized object, while explaining the
pose-difference between the two inputs. An equivariant representation vector
has two components: the invariant identity part, and a discernable encoding of
the transformation. Models that can't explain pose-differences risk "diluting"
the representation, in pursuit of optimizing a classification or regression
loss function.
  We introduce a Group Convolutional Neural Network with linear equivariance to
translations and right angle rotations in three dimensions. We call this
network CubeNet, reflecting its cube-like symmetry. By construction, this
network helps preserve a 3D shape's global and local signature, as it is
transformed through successive layers. We apply this network to a variety of 3D
inference problems, achieving state-of-the-art on the ModelNet10 classification
challenge, and comparable performance on the ISBI 2012 Connectome Segmentation
Benchmark. To the best of our knowledge, this is the first 3D rotation
equivariant CNN for voxel representations.
