# Score-based Generative Neural Networks for Large-Scale Optimal Transport

arXiv: https://arxiv.org/abs/2110.03237 · HTML: https://mlsanity.com/paper/2110.03237/

## Key facts

- Cited by 60 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 26 references that are in the map.
- First posted on arXiv 7 Oct 2021.

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

- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [An introduction to domain adaptation and transfer learning](https://mlsanity.com/paper/1812.11806/) (2018)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Adversarial score matching and improved sampling for image generation](https://mlsanity.com/paper/2009.05475/) (2020)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [Width Provably Matters in Optimization for Deep Linear Neural Networks](https://mlsanity.com/paper/1901.08572/) (2019)
- [Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks](https://mlsanity.com/paper/2007.04462/) (2020)
- [Continuous Regularized Wasserstein Barycenters](https://mlsanity.com/paper/2008.12534/) (2020)
- [Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm](https://mlsanity.com/paper/1905.13194/) (2019)
- [Adversarial Computation of Optimal Transport Maps](https://mlsanity.com/paper/1906.09691/) (2019)

## Built on by (most cited first, 30 of 60)

- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [TrojDiff: Trojan Attacks on Diffusion Models with Diverse Targets](https://mlsanity.com/paper/2303.05762/) (2023)
- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)
- [Guidance with Spherical Gaussian Constraint for Conditional Diffusion](https://mlsanity.com/paper/2402.03201/) (2024)
- [Dif-Fusion: Towards High Color Fidelity in Infrared and Visible Image Fusion with Diffusion Models](https://mlsanity.com/paper/2301.08072/) (2023)
- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [Light and Optimal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2402.03207/) (2024)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [Curriculum Direct Preference Optimization for Diffusion and Consistency Models](https://mlsanity.com/paper/2405.13637/) (2024)
- [A Denoising Diffusion Model for Fluid Field Prediction](https://mlsanity.com/paper/2301.11661/) (2023)
- [Attacks and Defenses for Generative Diffusion Models: A Comprehensive Survey](https://mlsanity.com/paper/2408.03400/) (2024)
- [Kernel Neural Optimal Transport](https://mlsanity.com/paper/2205.15269/) (2022)
- [SpectralDiff: A Generative Framework for Hyperspectral Image Classification with Diffusion Models](https://mlsanity.com/paper/2304.05961/) (2023)
- [GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomics](https://mlsanity.com/paper/2310.09254/) (2023)
- [Diffusion Model for Camouflaged Object Detection](https://mlsanity.com/paper/2308.00303/) (2023)
- [Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement](https://mlsanity.com/paper/2408.06277/) (2024)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Kantorovich Strikes Back! Wasserstein GANs are not Optimal Transport?](https://mlsanity.com/paper/2206.07767/) (2022)
- [Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmark](https://mlsanity.com/paper/2306.10161/) (2023)
- [Light Schr"odinger Bridge](https://mlsanity.com/paper/2310.01174/) (2023)
- [Human-Object Interaction Detection Collaborated with Large Relation-driven Diffusion Models](https://mlsanity.com/paper/2410.20155/) (2024)
- [Estimating Barycenters of Distributions with Neural Optimal Transport](https://mlsanity.com/paper/2402.03828/) (2024)
- [NoiseDiffusion: Correcting Noise for Image Interpolation with Diffusion Models beyond Spherical Linear Interpolation](https://mlsanity.com/paper/2403.08840/) (2024)
- [Adversarial Schr"odinger Bridge Matching](https://mlsanity.com/paper/2405.14449/) (2024)
- [Extremal Domain Translation with Neural Optimal Transport](https://mlsanity.com/paper/2301.12874/) (2023)
- [Energy-Guided Continuous Entropic Barycenter Estimation for General Costs](https://mlsanity.com/paper/2310.01105/) (2023)
- [Optimal Transport-Guided Conditional Score-Based Diffusion Models](https://mlsanity.com/paper/2311.01226/) (2023)

## Abstract (arXiv)

We consider the fundamental problem of sampling the optimal transport coupling between given source and target distributions. In certain cases, the optimal transport plan takes the form of a one-to-one mapping from the source support to the target support, but learning or even approximating such a map is computationally challenging for large and high-dimensional datasets due to the high cost of linear programming routines and an intrinsic curse of dimensionality. We study instead the Sinkhorn problem, a regularized form of optimal transport whose solutions are couplings between the source and the target distribution. We introduce a novel framework for learning the Sinkhorn coupling between two distributions in the form of a score-based generative model. Conditioned on source data, our procedure iterates Langevin Dynamics to sample target data according to the regularized optimal coupling. Key to this approach is a neural network parametrization of the Sinkhorn problem, and we prove convergence of gradient descent with respect to network parameters in this formulation. We demonstrate its empirical success on a variety of large scale optimal transport tasks.