# Learning representations that are closed-form Monge mapping optimal with application to domain adaptation

arXiv: https://arxiv.org/abs/2305.07500 · HTML: https://mlsanity.com/paper/2305.07500/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 70 references that are in the map.
- First posted on arXiv 12 May 2023.

## Builds on (most cited first, 30 of 37)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Deep CORAL: Correlation Alignment for Deep Domain Adaptation](https://mlsanity.com/paper/1607.01719/) (2016)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [VisDA: The Visual Domain Adaptation Challenge](https://mlsanity.com/paper/1710.06924/) (2017)
- [DeCAF: A Deep Convolutional Activation Feature for Generic Visual Recognition](https://mlsanity.com/paper/1310.1531/) (2013)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [A Survey of Unsupervised Deep Domain Adaptation](https://mlsanity.com/paper/1812.02849/) (2018)
- [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)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Style Transfer by Relaxed Optimal Transport and Self-Similarity](https://mlsanity.com/paper/1904.12785/) (2019)
- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018)
- [Theoretical Analysis of Domain Adaptation with Optimal Transport](https://mlsanity.com/paper/1610.04420/) (2016)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Optimal Transport for Multi-source Domain Adaptation under Target Shift](https://mlsanity.com/paper/1803.04899/) (2018)
- [Regularized Discrete Optimal Transport](https://mlsanity.com/paper/1307.5551/) (2013)
- [Learning Generative Models across Incomparable Spaces](https://mlsanity.com/paper/1905.05461/) (2019)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [Learning Wasserstein Embeddings](https://mlsanity.com/paper/1710.07457/) (2017)
- [Wasserstein Style Transfer](https://mlsanity.com/paper/1905.12828/) (2019)
- [Optimal Transport for Conditional Domain Matching and Label Shift](https://mlsanity.com/paper/2006.08161/) (2020)

## Built on by (most cited first, 2 of 2)

- [Optimal Transport for Domain Adaptation through Gaussian Mixture Models](https://mlsanity.com/paper/2403.13847/) (2024)
- [A Biconvex Formulation for Stable Transport of Mixture Models with a Unique Solution](https://mlsanity.com/paper/2606.02515/) (2026)

## Abstract (arXiv)

Optimal transport (OT) is a powerful geometric tool used to compare and align probability measures following the least effort principle. Despite its widespread use in machine learning (ML), OT problem still bears its computational burden, while at the same time suffering from the curse of dimensionality for measures supported on general high-dimensional spaces. In this paper, we propose to tackle these challenges using representation learning. In particular, we seek to learn an embedding space such that the samples of the two input measures become alignable in it with a simple affine mapping that can be calculated efficiently in closed-form. We then show that such approach leads to results that are comparable to solving the original OT problem when applied to the transfer learning task on which many OT baselines where previously evaluated in both homogeneous and heterogeneous DA settings. The code for our contribution is available at urlhttps://github.com/Oleffa/LaOT.