R${}^2$: Geometric Regressors Are Effective Point Cloud Registration Learners

We propose R${}^2$ a point cloud registration framework that recasts registration as one-shot shared geometry regression. Instead of learning explicit correspondences or descriptors, R${}^2$ predicts a shared, pair-conditioned canonical space for both partial observations. Using a compact set of geometric anchors, R${}^2$ assigns canonical coordinates to sampled points from both views, including points outside the overlapping regions. Correspondences are then derived naturally from the predicted canonical geometry through mutual nearest-neighbor matching, and the rigid transformation is recovered in the original Euclidean space using a closed-form SVD-based Kabsch solver. To improve canonicalization under partial overlap, we introduce a cross-view geometric coupling loss that complements the coordinate regression loss with relational supervision. The resulting framework requires only pairwise pose supervision to construct canonical targets and does not rely on annotated point-wise correspondence labels. Experiments on 3DMatch and 3DLoMatch demonstrate that R${}^2$ achieves competitive registration accuracy with a simple RANSAC-free inference pipeline and favorable speed–accuracy trade-offs.