# Neural Optimal Transport with General Cost Functionals

arXiv: https://arxiv.org/abs/2205.15403 · HTML: https://mlsanity.com/paper/2205.15403/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 60 of its 96 references that are in the map.
- First posted on arXiv 30 May 2022.

## Builds on (most cited first, 30 of 60)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [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)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Unsupervised Domain Adaptation by Backpropagation](https://mlsanity.com/paper/1409.7495/) (2014)
- [Learning Transferable Features with Deep Adaptation Networks](https://mlsanity.com/paper/1502.02791/) (2015)
- [CyCADA: Cycle-Consistent Adversarial Domain Adaptation](https://mlsanity.com/paper/1711.03213/) (2017)
- [Multimodal Unsupervised Image-to-Image Translation](https://mlsanity.com/paper/1804.04732/) (2018)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Deep Visual Domain Adaptation: A Survey](https://mlsanity.com/paper/1802.03601/) (2018)
- [MaskGAN: Towards Diverse and Interactive Facial Image Manipulation](https://mlsanity.com/paper/1907.11922/) (2019)
- [Deep Transfer Learning with Joint Adaptation Networks](https://mlsanity.com/paper/1605.06636/) (2016)
- [Unsupervised Image-to-Image Translation Networks](https://mlsanity.com/paper/1703.00848/) (2017)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Adversarial Autoencoders](https://mlsanity.com/paper/1511.05644/) (2015)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [DeepJDOT: Deep Joint Distribution Optimal Transport for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1803.10081/) (2018)
- [Dual Diffusion Implicit Bridges for Image-to-Image Translation](https://mlsanity.com/paper/2203.08382/) (2022)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Geometric Dataset Distances via Optimal Transport](https://mlsanity.com/paper/2002.02923/) (2020)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)

## Built on by (most cited first, 30 of 36)

- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)
- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [The Monge Gap: A Regularizer to Learn All Transport Maps](https://mlsanity.com/paper/2302.04953/) (2023)
- [On amortizing convex conjugates for optimal transport](https://mlsanity.com/paper/2210.12153/) (2022)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [Kantorovich Strikes Back! Wasserstein GANs are not Optimal Transport?](https://mlsanity.com/paper/2206.07767/) (2022)
- [Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmark](https://mlsanity.com/paper/2306.10161/) (2023)
- [Light Schr"odinger Bridge](https://mlsanity.com/paper/2310.01174/) (2023)
- [Estimating Barycenters of Distributions with Neural Optimal Transport](https://mlsanity.com/paper/2402.03828/) (2024)
- [Extremal Domain Translation with Neural Optimal Transport](https://mlsanity.com/paper/2301.12874/) (2023)
- [Energy-Guided Continuous Entropic Barycenter Estimation for General Costs](https://mlsanity.com/paper/2310.01105/) (2023)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Rethinking Optimal Transport in Offline Reinforcement Learning](https://mlsanity.com/paper/2410.14069/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Light Unbalanced Optimal Transport](https://mlsanity.com/paper/2303.07988/) (2023)
- [Uncovering Challenges of Solving the Continuous Gromov-Wasserstein Problem](https://mlsanity.com/paper/2303.05978/) (2023)
- [Taming Flow Matching with Unbalanced Optimal Transport into Fast Pansharpening](https://mlsanity.com/paper/2503.14975/) (2025)
- [Generating Synthetic Datasets by Interpolating along Generalized Geodesics](https://mlsanity.com/paper/2306.06866/) (2023)
- [Optimal Transport Maps are Good Voice Converters](https://mlsanity.com/paper/2411.02402/) (2024)
- [Improved Stochastic Optimization of LogSumExp](https://mlsanity.com/paper/2509.24894/) (2025)
- [Semi-supervised Learning of Pushforwards For Domain Translation & Adaptation](https://mlsanity.com/paper/2304.08673/) (2023)
- [OT-Net: A Reusable Neural Optimal Transport Solver](https://mlsanity.com/paper/2306.08233/) (2023)
- [A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers](https://mlsanity.com/paper/2502.01310/) (2025)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [Displacement-Sparse Neural Optimal Transport](https://mlsanity.com/paper/2502.01889/) (2025)
- [A Physics-preserved Transfer Learning Method for Differential Equations](https://mlsanity.com/paper/2505.01281/) (2025)
- [Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport](https://mlsanity.com/paper/2510.01153/) (2025)
- [Expert or not? assessing data quality in offline reinforcement learning](https://mlsanity.com/paper/2510.12638/) (2025)
- [Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization](https://mlsanity.com/paper/2410.02628/) (2024)

## Abstract (arXiv)

We introduce a novel neural network-based algorithm to compute optimal transport (OT) plans for general cost functionals. In contrast to common Euclidean costs, i.e., ℓ1 or ℓ2, such functionals provide more flexibility and allow using auxiliary information, such as class labels, to construct the required transport map. Existing methods for general costs are discrete and have limitations in practice, i.e. they do not provide an out-of-sample estimation. We address the challenge of designing a continuous OT approach for general costs that generalizes to new data points in high-dimensional spaces, such as images. Additionally, we provide the theoretical error analysis for our recovered transport plans. As an application, we construct a cost functional to map data distributions while preserving the class-wise structure.