Scalable and Differentiable Point-Cloud Registration Using Maximum Mean Discrepancy

We present MMD-Reg, a novel correspondence-free approach to point-cloud registration that is differentiable and has linear computational complexity in the number of points. We model registration as a nonlinear least-squares problem based on the Maximum Mean Discrepancy, approximated using random Fourier features. The resulting objective can be solved efficiently with standard methods such as Levenberg–Marquardt, and the solution is differentiable via the implicit function theorem. This allows MMD-Reg to be used as a differentiable optimization layer within end-to-end trainable models, supporting registration under challenging conditions such as poor initial alignment and partial overlap. We demonstrate this Neural MMD-Reg formulation by integrating the layer with a set transformer, training the resulting model in supervised and unsupervised settings, and comparing its performance against recent learning-based methods. We also evaluate standalone MMD-Reg, comparing its accuracy and scalability against widely used non-learning-based registration methods.

Aligning two 3D point clouds is an important problem in robotics and computer vision, though it becomes harder when scans are noisy or very large. Many existing methods try to match points directly, which can make them fragile and difficult to combine with modern deep learning systems. We introduce MMD-Reg, a new method that aligns point clouds by comparing their overall spatial distributions instead of matching individual points one by one. It is based on Maximum Mean Discrepancy (MMD), a statistical way to compare underlying data distributions. To make this practical for large point clouds, we use an efficient approximation based on random Fourier features, which allows the method to scale linearly with the number of points for a fixed feature dimension. Our approach is also differentiable, so it can be used both as a standalone registration method and as part of a trainable neural network. We show that it works well in both roles, across synthetic data and real outdoor LiDAR scans.