  A wide range of techniques have been proposed in recent years for designing
neural networks for 3D data that are equivariant under rotation and translation
of the input. Most approaches for equivariance under the Euclidean group
$\mathrm{SE}(3)$ of rotations and translations fall within one of the two major
categories. The first category consists of methods that use
$\mathrm{SE}(3)$-convolution which generalizes classical
$\mathbb{R}^3$-convolution on signals over $\mathrm{SE}(3)$. Alternatively, it
is possible to use \textit{steerable convolution} which achieves
$\mathrm{SE}(3)$-equivariance by imposing constraints on
$\mathbb{R}^3$-convolution of tensor fields. It is known by specialists in the
field that the two approaches are equivalent, with steerable convolution being
the Fourier transform of $\mathrm{SE}(3)$ convolution. Unfortunately, these
results are not widely known and moreover the exact relations between deep
learning architectures built upon these two approaches have not been precisely
described in the literature on equivariant deep learning. In this work we
provide an in-depth analysis of both methods and their equivalence and relate
the two constructions to multiview convolutional networks. Furthermore, we
provide theoretical justifications of separability of $\mathrm{SE}(3)$ group
convolution, which explain the applicability and success of some recent
approaches. Finally, we express different methods using a single coherent
formalism and provide explicit formulas that relate the kernels learned by
different methods. In this way, our work helps to unify different
previously-proposed techniques for achieving roto-translational equivariance,
and helps to shed light on both the utility and precise differences between
various alternatives. We also derive new TFN non-linearities from our
equivalence principle and test them on practical benchmark datasets.
