# Consistent Optimal Transport with Empirical Conditional Measures

arXiv: https://arxiv.org/abs/2305.15901 · HTML: https://mlsanity.com/paper/2305.15901/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 55 references that are in the map.
- First posted on arXiv 25 May 2023.

## Builds on (most cited first, 24 of 24)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Learning to Prompt for Vision-Language Models](https://mlsanity.com/paper/2109.01134/) (2021)
- [Enriching Word Vectors with Subword Information](https://mlsanity.com/paper/1607.04606/) (2016)
- [EuroSAT: A Novel Dataset and Deep Learning Benchmark for Land Use and Land Cover Classification](https://mlsanity.com/paper/1709.00029/) (2017)
- [Conditional Prompt Learning for Vision-Language Models](https://mlsanity.com/paper/2203.05557/) (2022)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Tip-Adapter: Training-free Adaption of CLIP for Few-shot Classification](https://mlsanity.com/paper/2207.09519/) (2022)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1709.01953/) (2017)
- [Learning with minibatch Wasserstein : asymptotic and gradient properties](https://mlsanity.com/paper/1910.04091/) (2019)
- [PLOT: Prompt Learning with Optimal Transport for Vision-Language Models](https://mlsanity.com/paper/2210.01253/) (2022)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Sinkhorn Divergences for Unbalanced Optimal Transport](https://mlsanity.com/paper/1910.12958/) (2019)
- [Conditional Bures Metric for Domain Adaptation](https://mlsanity.com/paper/2108.00302/) (2021)
- [Atlantic Causal Inference Conference (ACIC) Data Analysis Challenge 2017](https://mlsanity.com/paper/1905.09515/) (2019)
- [Wasserstein Generative Learning of Conditional Distribution](https://mlsanity.com/paper/2112.10039/) (2021)
- [Data Driven Conditional Optimal Transport](https://mlsanity.com/paper/1910.11422/) (2019)
- [Efficient Robust Optimal Transport with Application to Multi-Label Classification](https://mlsanity.com/paper/2010.11852/) (2020)
- [Training Lipschitz continuous operators using reproducing kernels](https://mlsanity.com/paper/2112.03344/) (2021)

## Built on by (most cited first, 6 of 6)

- [Submodular Framework for Structured-Sparse Optimal Transport](https://mlsanity.com/paper/2406.04914/) (2024)
- [Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances](https://mlsanity.com/paper/2310.13433/) (2023)
- [Pairwise Optimal Transports for Training All-to-All Flow-Based Condition Transfer Model](https://mlsanity.com/paper/2504.03188/) (2025)
- [Unsupervised Structural-Counterfactual Generation under Domain Shift](https://mlsanity.com/paper/2502.12013/) (2025)
- [Learning Counterfactually Fair Models via Improved Generation with Neural Causal Models](https://mlsanity.com/paper/2502.12796/) (2025)
- [Approximating Measures on Function Spaces: Transport and Truncation](https://mlsanity.com/paper/2609.17802/) (2026)

## Abstract (arXiv)

Given samples from two joint distributions, we consider the problem of Optimal Transportation (OT) between them when conditioned on a common variable. We focus on the general setting where the conditioned variable may be continuous, and the marginals of this variable in the two joint distributions may not be the same. In such settings, standard OT variants cannot be employed, and novel estimation techniques are necessary. Since the main challenge is that the conditional distributions are not explicitly available, the key idea in our OT formulation is to employ kernelized-least-squares terms computed over the joint samples, which implicitly match the transport plan's marginals with the empirical conditionals. Under mild conditions, we prove that our estimated transport plans, as a function of the conditioned variable, are asymptotically optimal. For finite samples, we show that the deviation in terms of our regularized objective is bounded by O(1/m1/4), where m is the number of samples. We also discuss how the conditional transport plan could be modelled using explicit probabilistic models as well as using implicit generative ones. We empirically verify the consistency of our estimator on synthetic datasets, where the optimal plan is analytically known. When employed in applications like prompt learning for few-shot classification and conditional-generation in the context of predicting cell responses to treatment, our methodology improves upon state-of-the-art methods.