# Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances

arXiv: https://arxiv.org/abs/2310.13433 · HTML: https://mlsanity.com/paper/2310.13433/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 40 references that are in the map.
- First posted on arXiv 20 Oct 2023.

## Builds on (most cited first, 23 of 23)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Guided Image Generation with Conditional Invertible Neural Networks](https://mlsanity.com/paper/1907.02392/) (2019)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint](https://mlsanity.com/paper/2109.11375/) (2021)
- [Sobolev GAN](https://mlsanity.com/paper/1711.04894/) (2017)
- [Benchmarking Invertible Architectures on Inverse Problems](https://mlsanity.com/paper/2101.10763/) (2021)
- [Collapse by Conditioning: Training Class-conditional GANs with Limited Data](https://mlsanity.com/paper/2201.06578/) (2022)
- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [Wasserstein Generative Learning of Conditional Distribution](https://mlsanity.com/paper/2112.10039/) (2021)
- [Generalized Normalizing Flows via Markov Chains](https://mlsanity.com/paper/2111.12506/) (2021)
- [Optimal Transport for Conditional Domain Matching and Label Shift](https://mlsanity.com/paper/2006.08161/) (2020)
- [Nonparametric Generative Modeling with Conditional Sliced-Wasserstein Flows](https://mlsanity.com/paper/2305.02164/) (2023)
- [Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems](https://mlsanity.com/paper/2303.15845/) (2023)
- [Data Driven Conditional Optimal Transport](https://mlsanity.com/paper/1910.11422/) (2019)
- [Solution of physics-based inverse problems using conditional generative adversarial networks with full gradient penalty](https://mlsanity.com/paper/2306.04895/) (2023)
- [Consistent Optimal Transport with Empirical Conditional Measures](https://mlsanity.com/paper/2305.15901/) (2023)
- [Wasserstein Geodesic Generator for Conditional Distributions](https://mlsanity.com/paper/2308.10145/) (2023)
- [About exchanging expectation and supremum for conditional Wasserstein GANs](https://mlsanity.com/paper/2103.13906/) (2021)
- [Data Interpolants -- That's What Discriminators in Higher-order Gradient-regularized GANs Are](https://mlsanity.com/paper/2306.00785/) (2023)

## Built on by (most cited first, 3 of 3)

- [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [Contextual Scenario Generation for Two-Stage Stochastic Programming](https://mlsanity.com/paper/2502.05349/) (2025)

## Abstract (arXiv)

In inverse problems, many conditional generative models approximate the posterior measure by minimizing a distance between the joint measure and its learned approximation. While this approach also controls the distance between the posterior measures in the case of the Kullback Leibler divergence, it does not hold true for the Wasserstein distance. We will introduce a conditional Wasserstein distance with a set of restricted couplings that equals the expected Wasserstein distance of the posteriors. By deriving its dual, we find a rigorous way to motivate the loss of conditional Wasserstein GANs. We outline conditions under which the vanilla and the conditional Wasserstein distance coincide. Furthermore, we will show numerical examples where training with the conditional Wasserstein distance yields favorable properties for posterior sampling.