Toward bilipshiz geometric models
Yonatan Sverdlov, Eitan Rosen, Nadav Dym
TL;DR
The paper investigates whether neural networks for point sets can preserve natural symmetry-aware distances under bi-Lipschitz guarantees. By comparing the Procrustes Matching metric and the Hard-Gromov-Wasserstein distance, it shows these metrics are not bi-Lipschitz equivalent, implying invariant networks cannot be bi-Lipschitz with respect to both. The authors develop bi-Lipschitz point-set models, including G_+ invariant constructions and higher-order WL-like embeddings, and provide theoretical guarantees alongside preliminary 2D/3D experiments demonstrating improved correspondence tasks over standard invariant models. The work advances understanding of when and how bi-Lipschitz guarantees can be achieved in symmetry-aware geometric learning, with practical implications for robust point-cloud matching and correspondences.
Abstract
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.