# Displacement-Sparse Neural Optimal Transport

arXiv: https://arxiv.org/abs/2502.01889 · HTML: https://mlsanity.com/paper/2502.01889/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 53 references that are in the map.
- First posted on arXiv 3 Feb 2025.

## Builds on (most cited first, 25 of 25)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networks](https://mlsanity.com/paper/2106.00774/) (2021)
- [Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks](https://mlsanity.com/paper/2007.04462/) (2020)
- [Improving the Improved Training of Wasserstein GANs: A Consistency Term and Its Dual Effect](https://mlsanity.com/paper/1803.01541/) (2018)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [Improved Algorithms for Convex-Concave Minimax Optimization](https://mlsanity.com/paper/2006.06359/) (2020)
- [2-Wasserstein Approximation via Restricted Convex Potentials with Application to Improved Training for GANs](https://mlsanity.com/paper/1902.07197/) (2019)
- [Sparsity-Constrained Optimal Transport](https://mlsanity.com/paper/2209.15466/) (2022)
- [The Monge Gap: A Regularizer to Learn All Transport Maps](https://mlsanity.com/paper/2302.04953/) (2023)
- [Learning Elastic Costs to Shape Monge Displacements](https://mlsanity.com/paper/2306.11895/) (2023)
- [ENOT: Expectile Regularization for Fast and Accurate Training of Neural Optimal Transport](https://mlsanity.com/paper/2403.03777/) (2024)
- [Optimal transport with f-divergence regularization and generalized Sinkhorn algorithm](https://mlsanity.com/paper/2105.14337/) (2021)
- [Sparsity of Quadratically Regularized Optimal Transport: Scalar Case](https://mlsanity.com/paper/2410.03353/) (2024)
- [Fast and scalable Wasserstein-1 neural optimal transport solver for single-cell perturbation prediction](https://mlsanity.com/paper/2411.00614/) (2024)
- [Quadratically Regularized Optimal Transport: Existence and Multiplicity of Potentials](https://mlsanity.com/paper/2404.06847/) (2024)
- [Manifold Learning with Sparse Regularised Optimal Transport](https://mlsanity.com/paper/2307.09816/) (2023)

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

- [Simplifying Optimal Transport through Schatten-p Regularization](https://mlsanity.com/paper/2510.11910/) (2025)

## Abstract (arXiv)

Optimal transport (OT) aims to find a map T that transports mass from one probability measure to another while minimizing a cost function. Recently, neural OT solvers have gained popularity in high dimensional biological applications such as drug perturbation, due to their superior computational and memory efficiency compared to traditional exact Sinkhorn solvers. However, the overly complex high dimensional maps learned by neural OT solvers often suffer from poor interpretability. Prior work addressed this issue in the context of exact OT solvers by introducing displacement-sparse maps via designed elastic cost, but such method failed to be applied to neural OT settings. In this work, we propose an intuitive and theoretically grounded approach to learning displacement-sparse maps within neural OT solvers. Building on our new formulation, we introduce a novel smoothed ℓ0 regularizer that outperforms the ℓ1 based alternative from prior work. Leveraging Input Convex Neural Network's flexibility, we further develop a heuristic framework for adaptively controlling sparsity intensity, an approach uniquely enabled by the neural OT paradigm. We demonstrate the necessity of this adaptive framework in large-scale, high-dimensional training, showing not only improved accuracy but also practical ease of use for downstream applications.