# Stochastic interpolants with data-dependent couplings

arXiv: https://arxiv.org/abs/2310.03725 · HTML: https://mlsanity.com/paper/2310.03725/

## Key facts

- Cited by 100 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 47 references that are in the map.
- First posted on arXiv 5 Oct 2023.

## Builds on (most cited first, 30 of 36)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (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)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Deep Networks with Stochastic Depth](https://mlsanity.com/paper/1603.09382/) (2016)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [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)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [Structure-based Drug Design with Equivariant Diffusion Models](https://mlsanity.com/paper/2210.13695/) (2022)

## Built on by (most cited first, 30 of 100)

- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [PaperBench: Evaluating AI's Ability to Replicate AI Research](https://mlsanity.com/paper/2504.01848/) (2025)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [DepthFM: Fast Monocular Depth Estimation with Flow Matching](https://mlsanity.com/paper/2403.13788/) (2024)
- [Any-Order Flexible Length Masked Diffusion](https://mlsanity.com/paper/2509.01025/) (2025)
- [Boosting Latent Diffusion with Flow Matching](https://mlsanity.com/paper/2312.07360/) (2023)
- [Improved motif-scaffolding with SE(3) flow matching](https://mlsanity.com/paper/2401.04082/) (2024)
- [Flow Matching with Gaussian Process Priors for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2410.03024/) (2024)
- [Meta Flow Maps enable scalable reward alignment](https://mlsanity.com/paper/2601.14430/) (2026)
- [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024)
- [Much Ado About Noising: Dispelling the Myths of Generative Robotic Control](https://mlsanity.com/paper/2512.01809/) (2025)
- [Posterior-Mean Rectified Flow: Towards Minimum MSE Photo-Realistic Image Restoration](https://mlsanity.com/paper/2410.00418/) (2024)
- [Flow matching achieves almost minimax optimal convergence](https://mlsanity.com/paper/2405.20879/) (2024)
- [Open Materials Generation with Stochastic Interpolants](https://mlsanity.com/paper/2502.02582/) (2025)
- [Flowing from Words to Pixels: A Noise-Free Framework for Cross-Modality Evolution](https://mlsanity.com/paper/2412.15213/) (2024)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [Thermalizer: Stable autoregressive neural emulation of spatiotemporal chaos](https://mlsanity.com/paper/2503.18731/) (2025)
- [FrameBridge: Improving Image-to-Video Generation with Bridge Models](https://mlsanity.com/paper/2410.15371/) (2024)
- [Gaussian Interpolation Flows](https://mlsanity.com/paper/2311.11475/) (2023)
- [EquiJump: Protein Dynamics Simulation via SO(3)-Equivariant Stochastic Interpolants](https://mlsanity.com/paper/2410.09667/) (2024)
- [VCT: Training Consistency Models with Variational Noise Coupling](https://mlsanity.com/paper/2502.18197/) (2025)
- [Flow Matching for Efficient and Scalable Data Assimilation](https://mlsanity.com/paper/2508.13313/) (2025)
- [All that structure matches does not glitter](https://mlsanity.com/paper/2509.12178/) (2025)
- [Extended Flow Matching: a Method of Conditional Generation with Generalized Continuity Equation](https://mlsanity.com/paper/2402.18839/) (2024)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [One-Step Offline Distillation of Diffusion-based Models via Koopman Modeling](https://mlsanity.com/paper/2505.13358/) (2025)
- [STFlow: Data-Coupled Flow Matching for Geometric Trajectory Simulation](https://mlsanity.com/paper/2505.18647/) (2025)
- [Learning Straight Flows: Variational Flow Matching for Efficient Generation](https://mlsanity.com/paper/2511.17583/) (2025)
- [LADB: Latent Aligned Diffusion Bridges for Semi-Supervised Domain Translation](https://mlsanity.com/paper/2509.08628/) (2025)
- [Stochastic Interpolants via Conditional Dependent Coupling](https://mlsanity.com/paper/2509.23122/) (2025)

## Abstract (arXiv)

Generative models inspired by dynamical transport of measure -- such as flows and diffusions -- construct a continuous-time map between two probability densities. Conventionally, one of these is the target density, only accessible through samples, while the other is taken as a simple base density that is data-agnostic. In this work, using the framework of stochastic interpolants, we formalize how to textitcouple the base and the target densities, whereby samples from the base are computed conditionally given samples from the target in a way that is different from (but does preclude) incorporating information about class labels or continuous embeddings. This enables us to construct dynamical transport maps that serve as conditional generative models. We show that these transport maps can be learned by solving a simple square loss regression problem analogous to the standard independent setting. We demonstrate the usefulness of constructing dependent couplings in practice through experiments in super-resolution and in-painting.