# DeepEMD: A Transformer-based Fast Estimation of the Earth Mover's Distance

arXiv: https://arxiv.org/abs/2311.09998 · HTML: https://mlsanity.com/paper/2311.09998/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 21 references that are in the map.
- First posted on arXiv 16 Nov 2023.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [Revisiting Point Cloud Classification: A New Benchmark Dataset and Classification Model on Real-World Data](https://mlsanity.com/paper/1908.04616/) (2019)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [3D Point Cloud Generative Adversarial Network Based on Tree Structured Graph Convolutions](https://mlsanity.com/paper/1905.06292/) (2019)
- [Density-aware Chamfer Distance as a Comprehensive Metric for Point Cloud Completion](https://mlsanity.com/paper/2111.12702/) (2021)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [SetVAE: Learning Hierarchical Composition for Generative Modeling of Set-Structured Data](https://mlsanity.com/paper/2103.15619/) (2021)
- [Meta Optimal Transport](https://mlsanity.com/paper/2206.05262/) (2022)
- [DPDist : Comparing Point Clouds Using Deep Point Cloud Distance](https://mlsanity.com/paper/2004.11784/) (2020)
- [InfoOT: Information Maximizing Optimal Transport](https://mlsanity.com/paper/2210.03164/) (2022)

## Built on by (most cited first, 2 of 2)

- [On permutation-invariant neural networks](https://mlsanity.com/paper/2403.17410/) (2024)
- [Deformable registration and generative modelling of aortic anatomies by auto-decoders and neural ODEs](https://mlsanity.com/paper/2506.00947/) (2025)

## Abstract (arXiv)

The Earth Mover's Distance (EMD) is the measure of choice between point clouds. However the computational cost to compute it makes it prohibitive as a training loss, and the standard approach is to use a surrogate such as the Chamfer distance. We propose an attention-based model to compute an accurate approximation of the EMD that can be used as a training loss for generative models. To get the necessary accurate estimation of the gradients we train our model to explicitly compute the matching between point clouds instead of EMD itself. We cast this new objective as the estimation of an attention matrix that approximates the ground truth matching matrix. Experiments show that this model provides an accurate estimate of the EMD and its gradient with a wall clock speed-up of more than two orders of magnitude with respect to the exact Hungarian matching algorithm and one order of magnitude with respect to the standard approximate Sinkhorn algorithm, allowing in particular to train a point cloud VAE with the EMD itself. Extensive evaluation show the remarkable behaviour of this model when operating out-of-distribution, a key requirement for a distance surrogate. Finally, the model generalizes very well to point clouds during inference several times larger than during training.