# Wasserstein Generative Learning of Conditional Distribution

arXiv: https://arxiv.org/abs/2112.10039 · HTML: https://mlsanity.com/paper/2112.10039/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 19 Dec 2021.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Nonlinear Approximation via Compositions](https://mlsanity.com/paper/1902.10170/) (2019)
- [Converting High-Dimensional Regression to High-Dimensional Conditional Density Estimation](https://mlsanity.com/paper/1704.08095/) (2017)
- [Conditional Density Estimation Tools in Python and R with Applications to Photometric Redshifts and Likelihood-Free Cosmological Inference](https://mlsanity.com/paper/1908.11523/) (2019)
- [Nonparametric Conditional Density Estimation in a High-Dimensional Regression Setting](https://mlsanity.com/paper/1604.00540/) (2016)
- [On the capacity of deep generative networks for approximating distributions](https://mlsanity.com/paper/2101.12353/) (2021)

## Built on by (most cited first, 24 of 24)

- [CARD: Classification and Regression Diffusion Models](https://mlsanity.com/paper/2206.07275/) (2022)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference](https://mlsanity.com/paper/2310.16975/) (2023)
- [Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems](https://mlsanity.com/paper/2303.15845/) (2023)
- [Wasserstein Generative Regression](https://mlsanity.com/paper/2306.15163/) (2023)
- [Learning from small data sets: Patch-based regularizers in inverse problems for image reconstruction](https://mlsanity.com/paper/2312.16611/) (2023)
- [A generative flow for conditional sampling via optimal transport](https://mlsanity.com/paper/2307.04102/) (2023)
- [A Kernel-Based Conditional Two-Sample Test Using Nearest Neighbors (with Applications to Calibration, Regression Curves, and Simulation-Based Inference)](https://mlsanity.com/paper/2407.16550/) (2024)
- [Consistent Optimal Transport with Empirical Conditional Measures](https://mlsanity.com/paper/2305.15901/) (2023)
- [A Likelihood Based Approach to Distribution Regression Using Conditional Deep Generative Models](https://mlsanity.com/paper/2410.02025/) (2024)
- [Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances](https://mlsanity.com/paper/2310.13433/) (2023)
- [DistPred: A Distribution-Free Probabilistic Inference Method for Regression and Forecasting](https://mlsanity.com/paper/2406.11397/) (2024)
- [Wasserstein Distributionally Robust Nonparametric Regression](https://mlsanity.com/paper/2505.07967/) (2025)
- [Non-Asymptotic Bounds for Adversarial Excess Risk under Misspecified Models](https://mlsanity.com/paper/2309.00771/) (2023)
- [Diffusion Boosted Trees](https://mlsanity.com/paper/2406.01813/) (2024)
- [Deep Limit Model-free Prediction in Regression](https://mlsanity.com/paper/2408.09532/) (2024)
- [Penalized Generative Variable Selection](https://mlsanity.com/paper/2402.16661/) (2024)
- [Deep Generative Demand Learning for Newsvendor and Pricing](https://mlsanity.com/paper/2411.08631/) (2024)
- [Learning Counterfactually Fair Models via Improved Generation with Neural Causal Models](https://mlsanity.com/paper/2502.12796/) (2025)
- [Boosting Statistic Learning with Synthetic Data from Pretrained Large Models](https://mlsanity.com/paper/2505.04992/) (2025)
- [Calibration Prediction Interval for Non-parametric Regression and Neural Networks](https://mlsanity.com/paper/2509.02735/) (2025)
- [One-shot Conditional Sampling: MMD meets Nearest Neighbors](https://mlsanity.com/paper/2509.25507/) (2025)
- [DenoiseRank: Learning to Rank by Diffusion Models](https://mlsanity.com/paper/2604.20852/) (2026)
- [Beyond Expected Information Gain: Stable Bayesian Optimal Experimental Design with Integral Probability Metrics and Plug-and-Play Extensions](https://mlsanity.com/paper/2604.21849/) (2026)

## Abstract (arXiv)

Conditional distribution is a fundamental quantity for describing the relationship between a response and a predictor. We propose a Wasserstein generative approach to learning a conditional distribution. The proposed approach uses a conditional generator to transform a known distribution to the target conditional distribution. The conditional generator is estimated by matching a joint distribution involving the conditional generator and the target joint distribution, using the Wasserstein distance as the discrepancy measure for these joint distributions. We establish non-asymptotic error bound of the conditional sampling distribution generated by the proposed method and show that it is able to mitigate the curse of dimensionality, assuming that the data distribution is supported on a lower-dimensional set. We conduct numerical experiments to validate proposed method and illustrate its applications to conditional sample generation, nonparametric conditional density estimation, prediction uncertainty quantification, bivariate response data, image reconstruction and image generation.