提出点云网络的双利普希茨保证，提升匹配精度。
Toward bilipshiz geometric models
- 用双利普希茨等价性分析点云对称性保持能力。
- 发现标准不变网络在质心匹配距离下不满足双利普希茨。
- 改进网络结构实现双利普希茨性，适合3D点云对应任务。
许多点云神经网络天生对排列和刚性变换具有不变性。本文探讨此类网络是否保持点云空间中自然的对称感知距离，通过双利普希茨等价性概念进行分析。研究聚焦于两种对称感知度量：(a) 质心匹配（Procrustes Matching, PM）距离，(b) 硬格罗莫夫-沃瑟斯坦（Hard Gromov Wasserstein）距离。我们证明这两种距离本身并非双利普希茨等价，从而推导出流行的点云不变网络在PM度量下不满足双利普希茨性质。随后，我们提出改进方法使网络获得双利普希茨保证。最后，初步实验表明，所提双利普希茨模型在3D点云对应任务中优于标准不变模型。
原文摘要 · Abstract (English)
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.
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