  The goal of this paper is to address the problem of \textit{global} point
cloud registration (PCR) i.e., finding the optimal alignment between point
clouds irrespective of the initial poses of the scans. This problem is
notoriously challenging for classical optimization methods due to computational
constraints. First, we show that state-of-the-art deep learning methods suffer
from huge performance degradation when the point clouds are arbitrarily placed
in space. We propose that \textit{equivariant deep learning} should be utilized
for solving this task and we characterize the specific type of bi-equivariance
of PCR. Then, we design BiEquiformer a novel and scalable
\textit{bi-equivariant} pipeline i.e. equivariant to the independent
transformations of the input point clouds. While a naive approach would process
the point clouds independently we design expressive bi-equivariant layers that
fuse the information from both point clouds. This allows us to extract
high-quality superpoint correspondences and in turn, robust point-cloud
registration. Extensive comparisons against state-of-the-art methods show that
our method achieves comparable performance in the canonical setting and
superior performance in the robust setting in both the 3DMatch and the
challenging low-overlap 3DLoMatch dataset.
