Toward bilipshiz geometric models
This addresses a theoretical gap in geometric deep learning for researchers, but it is incremental as it builds on existing equivariant learning concepts.
The paper tackles the problem of whether invariant neural networks for point clouds preserve symmetry-aware distances, showing they are not bi-Lipschitz with respect to the Procrustes Matching metric and modifying them to achieve bi-Lipschitz guarantees, with initial experiments demonstrating advantages in finding correspondences between 3D point clouds.
Many neural networks for point clouds are, by design, invariant to the symmetries of this datatype: permutations and rigid motions. The purpose of this paper is to examine whether such networks preserve natural symmetry aware distances on the point cloud spaces, through the notion of bi-Lipschitz equivalence. This inquiry is motivated by recent work in the Equivariant learning literature which highlights the advantages of bi-Lipschitz models in other scenarios. We consider two symmetry aware metrics on point clouds: (a) The Procrustes Matching (PM) metric and (b) Hard Gromov Wasserstien distances. We show that these two distances themselves are not bi-Lipschitz equivalent, and as a corollary deduce that popular invariant networks for point clouds are not bi-Lipschitz with respect to the PM metric. We then show how these networks can be modified so that they do obtain bi-Lipschitz guarantees. Finally, we provide initial experiments showing the advantage of the proposed bi-Lipschitz model over standard invariant models, for the tasks of finding correspondences between 3D point clouds.