  In many practical applications, 3D point cloud analysis requires rotation
invariance. In this paper, we present a learnable descriptor invariant under 3D
rotations and reflections, i.e., the O(3) actions, utilizing the recently
introduced steerable 3D spherical neurons and vector neurons. Specifically, we
propose an embedding of the 3D spherical neurons into 4D vector neurons, which
leverages end-to-end training of the model. In our approach, we perform
TetraTransform--an equivariant embedding of the 3D input into 4D, constructed
from the steerable neurons--and extract deeper O(3)-equivariant features using
vector neurons. This integration of the TetraTransform into the VN-DGCNN
framework, termed TetraSphere, negligibly increases the number of parameters by
less than 0.0002%. TetraSphere sets a new state-of-the-art performance
classifying randomly rotated real-world object scans of the challenging subsets
of ScanObjectNN. Additionally, TetraSphere outperforms all equivariant methods
on randomly rotated synthetic data: classifying objects from ModelNet40 and
segmenting parts of the ShapeNet shapes. Thus, our results reveal the practical
value of steerable 3D spherical neurons for learning in 3D Euclidean space. The
code is available at https://github.com/pavlo-melnyk/tetrasphere.
