# Toward bilipshiz geometric models

arXiv: https://arxiv.org/abs/2511.11735 · HTML: https://mlsanity.com/paper/2511.11735/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 47 references that are in the map.
- First posted on arXiv 13 Nov 2025.

## Builds on (most cited first, 29 of 29)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Deep Closest Point: Learning Representations for Point Cloud Registration](https://mlsanity.com/paper/1905.03304/) (2019)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Sign and Basis Invariant Networks for Spectral Graph Representation Learning](https://mlsanity.com/paper/2202.13013/) (2022)
- [On the equivalence between graph isomorphism testing and function approximation with GNNs](https://mlsanity.com/paper/1905.12560/) (2019)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [On the Universality of Rotation Equivariant Point Cloud Networks](https://mlsanity.com/paper/2010.02449/) (2020)
- [Low Dimensional Invariant Embeddings for Universal Geometric Learning](https://mlsanity.com/paper/2205.02956/) (2022)
- [Is Distance Matrix Enough for Geometric Deep Learning?](https://mlsanity.com/paper/2302.05743/) (2023)
- [Permutation Invariant Representations with Applications to Graph Deep Learning](https://mlsanity.com/paper/2203.07546/) (2022)
- [Group-invariant max filtering](https://mlsanity.com/paper/2205.14039/) (2022)
- [On the Completeness of Invariant Geometric Deep Learning Models](https://mlsanity.com/paper/2402.04836/) (2024)
- [Exact Recovery with Symmetries for Procrustes Matching](https://mlsanity.com/paper/1606.01548/) (2016)
- [Three iterations of (d-1)-WL test distinguish non isometric clouds of d-dimensional points](https://mlsanity.com/paper/2303.12853/) (2023)
- [Recognizing Rigid Patterns of Unlabeled Point Clouds by Complete and Continuous Isometry Invariants with no False Negatives and no False Positives](https://mlsanity.com/paper/2303.15385/) (2023)
- [FSW-GNN: A Bi-Lipschitz WL-Equivalent Graph Neural Network](https://mlsanity.com/paper/2410.09118/) (2024)
- [Fourier Sliced-Wasserstein Embedding for Multisets and Measures](https://mlsanity.com/paper/2504.02544/) (2025)
- [Linearly Converging Quasi Branch and Bound Algorithms for Global Rigid Registration](https://mlsanity.com/paper/1904.02204/) (2019)
- [ChiENN: Embracing Molecular Chirality with Graph Neural Networks](https://mlsanity.com/paper/2307.02198/) (2023)
- [Stability of sorting based embeddings](https://mlsanity.com/paper/2410.05446/) (2024)
- [Bi-Lipschitz Ansatz for Anti-Symmetric Functions](https://mlsanity.com/paper/2503.04263/) (2025)
- [The stability of generalized phase retrieval problem over compact groups](https://mlsanity.com/paper/2505.04190/) (2025)
- [Quantitative Bounds for Sorting-Based Permutation-Invariant Embeddings](https://mlsanity.com/paper/2510.22186/) (2025)

## Built on by (most cited first, 1 of 1)

- [Quantitative Approximation Rates for Group Equivariant Learning](https://mlsanity.com/paper/2602.20370/) (2026)

## Abstract (arXiv)

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.