# Schr"odinger bridge based deep conditional generative learning

arXiv: https://arxiv.org/abs/2409.17294 · HTML: https://mlsanity.com/paper/2409.17294/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 46 references that are in the map.
- First posted on arXiv 25 Sep 2024.

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

- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed](https://mlsanity.com/paper/2101.02388/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Conditional Image Generation with Score-Based Diffusion Models](https://mlsanity.com/paper/2111.13606/) (2021)
- [Learning Fast Samplers for Diffusion Models by Differentiating Through Sample Quality](https://mlsanity.com/paper/2202.05830/) (2022)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [ProDiff: Progressive Fast Diffusion Model For High-Quality Text-to-Speech](https://mlsanity.com/paper/2207.06389/) (2022)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [Bayesian Learning via Neural Schr"odinger-F"ollmer Flows](https://mlsanity.com/paper/2111.10510/) (2021)
- [Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference](https://mlsanity.com/paper/2006.06755/) (2020)
- [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)
- [Deep conditional distribution learning via conditional F"ollmer flow](https://mlsanity.com/paper/2402.01460/) (2024)

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

- [Finding Local Diffusion Schr"odinger Bridge using Kolmogorov-Arnold Network](https://mlsanity.com/paper/2502.19754/) (2025)
- [Deterministic Decomposition of Stochastic Generative Dynamics](https://mlsanity.com/paper/2605.08794/) (2026)
- [BridgeTwist: Twisting Schr"odinger Bridges for Training-Free Conditional Sampling](https://mlsanity.com/paper/2610.18658/) (2026)

## Abstract (arXiv)

Conditional generative models represent a significant advancement in the field of machine learning, allowing for the controlled synthesis of data by incorporating additional information into the generation process. In this work we introduce a novel Schr"odinger bridge based deep generative method for learning conditional distributions. We start from a unit-time diffusion process governed by a stochastic differential equation (SDE) that transforms a fixed point at time 0 into a desired target conditional distribution at time 1. For effective implementation, we discretize the SDE with Euler-Maruyama method where we estimate the drift term nonparametrically using a deep neural network. We apply our method to both low-dimensional and high-dimensional conditional generation problems. The numerical studies demonstrate that though our method does not directly provide the conditional density estimation, the samples generated by this method exhibit higher quality compared to those obtained by several existing methods. Moreover, the generated samples can be effectively utilized to estimate the conditional density and related statistical quantities, such as conditional mean and conditional standard deviation.