# Concentration bounds for linear Monge mapping estimation and optimal transport domain adaptation

arXiv: https://arxiv.org/abs/1905.10155 · HTML: https://mlsanity.com/paper/1905.10155/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2019.

## Built on by (most cited first, 17 of 17)

- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
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- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [Multi-Source and Test-Time Domain Adaptation on Multivariate Signals using Spatio-Temporal Monge Alignment](https://mlsanity.com/paper/2407.14303/) (2024)
- [Cycle Consistent Probability Divergences Across Different Spaces](https://mlsanity.com/paper/2111.11328/) (2021)
- [Input Convex Kolmogorov Arnold Networks](https://mlsanity.com/paper/2505.21208/) (2025)
- [Mitigating Source Bias for Fairer Weak Supervision](https://mlsanity.com/paper/2303.17713/) (2023)
- [Optimal Transport for Domain Adaptation through Gaussian Mixture Models](https://mlsanity.com/paper/2403.13847/) (2024)
- [Learning representations that are closed-form Monge mapping optimal with application to domain adaptation](https://mlsanity.com/paper/2305.07500/) (2023)
- [SKADA-Bench: Benchmarking Unsupervised Domain Adaptation Methods with Realistic Validation On Diverse Modalities](https://mlsanity.com/paper/2407.11676/) (2024)
- [Towards Backwards-Compatible Data with Confounded Domain Adaptation](https://mlsanity.com/paper/2203.12720/) (2022)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Convolutional Monge Mapping Normalization for learning on sleep data](https://mlsanity.com/paper/2305.18831/) (2023)
- [Predictive Coresets](https://mlsanity.com/paper/2502.05725/) (2025)
- [Optimal Transport with Heterogeneously Missing Data](https://mlsanity.com/paper/2505.17291/) (2025)
- [Convolutional Monge Mapping between EEG Datasets to Support Independent Component Labeling](https://mlsanity.com/paper/2509.01721/) (2025)

## Abstract (arXiv)

This article investigates the quality of the estimator of the linear Monge mapping between distributions. We provide the first concentration result on the linear mapping operator and prove a sample complexity of n-1/2 when using empirical estimates of first and second order moments. This result is then used to derive a generalization bound for domain adaptation with optimal transport. As a consequence, this method approaches the performance of theoretical Bayes predictor under mild conditions on the covariance structure of the problem. We also discuss the computational complexity of the linear mapping estimation and show that when the source and target are stationary the mapping is a convolution that can be estimated very efficiently using fast Fourier transforms. Numerical experiments reproduce the behavior of the proven bounds on simulated and real data for mapping estimation and domain adaptation on images.