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Even though considerable progress has been made in deep learning-based 3Dpoint cloud processing, how to obtain accurate correspondences for robustregistration remains a major challenge because existing hard assignment methodscannot deal with outliers naturally. Alternatively, the soft matching-basedmethods have been proposed to learn the matching probability rather than hardassignment. However, in this paper, we prove that these methods have aninherent ambiguity causing many deceptive correspondences. To address the abovechallenges, we propose to learn a partial permutation matching matrix, whichdoes not assign corresponding points to outliers, and implements hardassignment to prevent ambiguity. However, this proposal poses two new problems,i.e., existing hard assignment algorithms can only solve a full rankpermutation matrix rather than a partial permutation matrix, and this desiredmatrix is defined in the discrete space, which is non-differentiable. Inresponse, we design a dedicated soft-to-hard (S2H) matching procedure withinthe registration pipeline consisting of two steps: solving the soft matchingmatrix (S-step) and projecting this soft matrix to the partial permutationmatrix (H-step). Specifically, we augment the profit matrix before the hardassignment to solve an augmented permutation matrix, which is cropped toachieve the final partial permutation matrix. Moreover, to guarantee end-to-endlearning, we supervise the learned partial permutation matrix but propagate thegradient to the soft matrix instead. Our S2H matching procedure can be easilyintegrated with existing registration frameworks, which has been verified inrepresentative frameworks including DCP, RPMNet, and DGR. Extensive experimentshave validated our method, which creates a new state-of-the-art performance forrobust 3D point cloud registration. The code will be made public.

End-to-end Learning the Partial Permutation Matrix for Robust 3D Point Cloud Registration