  Given a pair of point clouds, the goal of assembly is to recover a rigid
transformation that aligns one point cloud to the other. This task is
challenging because the point clouds may be non-overlapped, and they may have
arbitrary initial positions. To address these difficulties, we propose a
method, called SE(3)-bi-equivariant transformer (BITR), based on the
SE(3)-bi-equivariance prior of the task: it guarantees that when the inputs are
rigidly perturbed, the output will transform accordingly. Due to its
equivariance property, BITR can not only handle non-overlapped PCs, but also
guarantee robustness against initial positions. Specifically, BITR first
extracts features of the inputs using a novel $SE(3) \times SE(3)$-transformer,
and then projects the learned feature to group SE(3) as the output. Moreover,
we theoretically show that swap and scale equivariances can be incorporated
into BITR, thus it further guarantees stable performance under scaling and
swapping the inputs. We experimentally show the effectiveness of BITR in
practical tasks.
