# Improving Neural Optimal Transport via Displacement Interpolation

arXiv: https://arxiv.org/abs/2410.03783 · HTML: https://mlsanity.com/paper/2410.03783/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 43 references that are in the map.
- First posted on arXiv 3 Oct 2024.

## Builds on (most cited first, 26 of 26)

- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Stabilizing Training of Generative Adversarial Networks through Regularization](https://mlsanity.com/paper/1705.09367/) (2017)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Diffusion Normalizing Flow](https://mlsanity.com/paper/2110.07579/) (2021)
- [Schr"odinger Bridge Flow for Unpaired Data Translation](https://mlsanity.com/paper/2409.09347/) (2024)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [Scalable Wasserstein Gradient Flow for Generative Modeling through Unbalanced Optimal Transport](https://mlsanity.com/paper/2402.05443/) (2024)
- [Kantorovich Strikes Back! Wasserstein GANs are not Optimal Transport?](https://mlsanity.com/paper/2206.07767/) (2022)
- [Estimating Barycenters of Distributions with Neural Optimal Transport](https://mlsanity.com/paper/2402.03828/) (2024)
- [Analyzing and Improving Optimal-Transport-based Adversarial Networks](https://mlsanity.com/paper/2310.02611/) (2023)
- [On the Uniqueness of Kantorovich Potentials](https://mlsanity.com/paper/2201.08316/) (2022)
- [Regularity theory and geometry of unbalanced optimal transport](https://mlsanity.com/paper/2112.11056/) (2021)

## Built on by (most cited first, 7 of 7)

- [Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport: A Smoothing Approach for Learning the Optimal Transport Plan](https://mlsanity.com/paper/2502.04583/) (2025)
- [Unpaired Image-to-Image Translation via a Self-Supervised Semantic Bridge](https://mlsanity.com/paper/2602.16664/) (2026)
- [Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport](https://mlsanity.com/paper/2510.01153/) (2025)
- [Implicit Neural Optimal Transport via Fixed-Point Optimization](https://mlsanity.com/paper/2605.10792/) (2026)
- [UOTIP: Unbalanced Optimal Transport Map for Unpaired Inverse Problems](https://mlsanity.com/paper/2605.21094/) (2026)
- [Source-Agnostic Image Translation Based on Latent Aware Adaptive Masking](https://mlsanity.com/paper/2608.14046/) (2026)
- [Deep Barycentric Regression for Optimal Transport Map Estimation and its Statistical Optimality](https://mlsanity.com/paper/2609.06598/) (2026)

## Abstract (arXiv)

Optimal Transport (OT) theory investigates the cost-minimizing transport map that moves a source distribution to a target distribution. Recently, several approaches have emerged for learning the optimal transport map for a given cost function using neural networks. We refer to these approaches as the OT Map. OT Map provides a powerful tool for diverse machine learning tasks, such as generative modeling and unpaired image-to-image translation. However, existing methods that utilize max-min optimization often experience training instability and sensitivity to hyperparameters. In this paper, we propose a novel method to improve stability and achieve a better approximation of the OT Map by exploiting displacement interpolation, dubbed Displacement Interpolation Optimal Transport Model (DIOTM). We derive the dual formulation of displacement interpolation at specific time t and prove how these dual problems are related across time. This result allows us to utilize the entire trajectory of displacement interpolation in learning the OT Map. Our method improves the training stability and achieves superior results in estimating optimal transport maps. We demonstrate that DIOTM outperforms existing OT-based models on image-to-image translation tasks.