# Inverse optimal transport

arXiv: https://arxiv.org/abs/1905.03950 · HTML: https://mlsanity.com/paper/1905.03950/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 May 2019.

## Built on by (most cited first, 26 of 26)

- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [A Unified Framework for Implicit Sinkhorn Differentiation](https://mlsanity.com/paper/2205.06688/) (2022)
- [Revisiting Deep Audio-Text Retrieval Through the Lens of Transportation](https://mlsanity.com/paper/2405.10084/) (2024)
- [Discrete Probabilistic Inverse Optimal Transport](https://mlsanity.com/paper/2112.09754/) (2021)
- [OTMatch: Improving Semi-Supervised Learning with Optimal Transport](https://mlsanity.com/paper/2310.17455/) (2023)
- [Double-Bounded Optimal Transport for Advanced Clustering and Classification](https://mlsanity.com/paper/2401.11418/) (2024)
- [Joint trajectory and network inference via reference fitting](https://mlsanity.com/paper/2409.06879/) (2024)
- [Computing Approximate Graph Edit Distance via Optimal Transport](https://mlsanity.com/paper/2412.18857/) (2024)
- [Your contrastive learning problem is secretly a distribution alignment problem](https://mlsanity.com/paper/2502.20141/) (2025)
- [Unsupervised Ground Metric Learning using Wasserstein Singular Vectors](https://mlsanity.com/paper/2102.06278/) (2021)
- [S-MolSearch: 3D Semi-supervised Contrastive Learning for Bioactive Molecule Search](https://mlsanity.com/paper/2409.07462/) (2024)
- [PhysioMTL: Personalizing Physiological Patterns using Optimal Transport Multi-Task Regression](https://mlsanity.com/paper/2203.12595/) (2022)
- [Learning non-equilibrium diffusions with Schr"odinger bridges: from exactly solvable to simulation-free](https://mlsanity.com/paper/2505.16644/) (2025)
- [Cracking Instance Jigsaw Puzzles: An Alternative to Multiple Instance Learning for Whole Slide Image Analysis](https://mlsanity.com/paper/2507.08178/) (2025)
- [Differentiable Cost-Parameterized Monge Map Estimators](https://mlsanity.com/paper/2406.08399/) (2024)
- [Optimal Transport for Brain-Image Alignment: Unveiling Redundancy and Synergy in Neural Information Processing](https://mlsanity.com/paper/2503.10663/) (2025)
- [Negative-Binomial Randomized Gamma Markov Processes for Heterogeneous Overdispersed Count Time Series](https://mlsanity.com/paper/2402.18995/) (2024)
- [Fast unsupervised ground metric learning with tree-Wasserstein distance](https://mlsanity.com/paper/2411.07432/) (2024)
- [Unsupervised Ground Metric Learning](https://mlsanity.com/paper/2507.13094/) (2025)
- [Enhancing 3D Medical Image Understanding with Pretraining Aided by 2D Multimodal Large Language Models](https://mlsanity.com/paper/2509.09064/) (2025)
- [Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization](https://mlsanity.com/paper/2410.02628/) (2024)
- [MELCOT: A Hybrid Learning Architecture with Marginal Preservation for Matrix-Valued Regression](https://mlsanity.com/paper/2509.23315/) (2025)
- [Reliable Modeling of Distribution Shifts via Displacement-Reshaped Optimal Transport](https://mlsanity.com/paper/2605.04965/) (2026)
- [Learning Urban Access Costs from Origin-Destination Flows via Inverse Optimal Transport](https://mlsanity.com/paper/2606.14157/) (2026)
- [Sinkhorn Linearization and the Spectral Proxy: Unifying the Statistical and Algorithmic Theory of Feature-Parameterized Inverse Optimal Transport via a Single Spectral Sandwich](https://mlsanity.com/paper/2608.13201/) (2026)
- [CloneOT: predicting clone-specific perturbation responses with lineage-supervised neural inverse optimal transport](https://mlsanity.com/paper/2610.16093/) (2026)

## Abstract (arXiv)

Discrete optimal transportation problems arise in various contexts in engineering, the sciences and the social sciences. Often the underlying cost criterion is unknown, or only partly known, and the observed optimal solutions are corrupted by noise. In this paper we propose a systematic approach to infer unknown costs from noisy observations of optimal transportation plans. The algorithm requires only the ability to solve the forward optimal transport problem, which is a linear program, and to generate random numbers. It has a Bayesian interpretation, and may also be viewed as a form of stochastic optimization. We illustrate the developed methodologies using the example of international migration flows. Reported migration flow data captures (noisily) the number of individuals moving from one country to another in a given period of time. It can be interpreted as a noisy observation of an optimal transportation map, with costs related to the geographical position of countries. We use a graph-based formulation of the problem, with countries at the nodes of graphs and non-zero weighted adjacencies only on edges between countries which share a border. We use the proposed algorithm to estimate the weights, which represent cost of transition, and to quantify uncertainty in these weights.