# Barron Optimal Transport I: Generative Modeling

arXiv: https://arxiv.org/abs/2610.10875 · HTML: https://mlsanity.com/paper/2610.10875/

## Key facts

- No citing papers in the map yet (data through 7 Oct 2026).
- Builds on 16 of its 82 references that are in the map.
- First posted on arXiv 7 Oct 2026.

## Builds on (most cited first, 16 of 16)

- [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)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [The Hidden Width of Deep ResNets: Tight Error Bounds and Phase Diagram](https://mlsanity.com/paper/2509.10167/) (2025)
- [Deep Learning as a Convex Paradigm of Computation: Minimizing Circuit Size with ResNets](https://mlsanity.com/paper/2511.20888/) (2025)
- [Benamou-Brenier Formulation of Optimal Transport for Nonlinear Control Systems on Rd](https://mlsanity.com/paper/2407.16088/) (2024)
- [Cryptographic Hardness of Score Estimation](https://mlsanity.com/paper/2404.03272/) (2024)
- [Distribution learning via neural differential equations: minimal energy regularization and approximation theory](https://mlsanity.com/paper/2502.03795/) (2025)
- [A Dynamical Systems Perspective on the Analysis of Neural Networks](https://mlsanity.com/paper/2507.05164/) (2025)
- [The Multiscale Single-Index Model: A Stylized Model for Hierarchical Feature Learning](https://mlsanity.com/paper/2607.03347/) (2026)
- [On the Hardness of Sampling from Mixture Distributions via Langevin Dynamics](https://mlsanity.com/paper/2406.02017/) (2024)
- [Learning Paths for Dynamic Measure Transport: A Control Perspective](https://mlsanity.com/paper/2511.03797/) (2025)
- [Radon--Wasserstein Gradient Flows for Interacting-Particle Sampling in High Dimensions](https://mlsanity.com/paper/2602.05227/) (2026)

## Abstract (arXiv)

Motivated by recent applications in generative modeling and sampling, we introduce a framework for optimal measure transport where cost captures the notion of neural network complexity. In transport-based generative models, samples from a reference distribution (e.g. Gaussian) are mapped to samples of a target distribution along ordinary or stochastic differential equations. These are implemented as deep residual networks when discretized in time, where each hidden layer approximates the associated instantaneous velocity. Thus, given a pair of target and reference measures, a natural question is to search for the most efficient neural representation that implements this transport. Our starting point is the kinetic formulation of OT, due to Benamou and Brenier. We replace the average kinetic L2 energy by the Barron energy citebach2017breaking, ma2022barron, a natural norm which measures the complexity of representing a given vector field with a neural hidden layer, and which captures the adaptive properties of feature learning. This defines a metric on the space of probability measures, complementing existing Wasserstein and Stein geometries. In this work we examine the properties of this metric in the context of generative modeling. As a first application, we quantify the suboptimality of diffusion generative modeling in the Barron geometry by establishing super-polynomial score approximation lower bounds for data generated by neural network pushforwards of the Gaussian. We then investigate the benefit of adaptivity as a way to study alternative generative models. In a companion paper citecompanionpaper we leverage the Barron transport geometry for sampling applications, extending the scope of Stein variational gradient methods via feature adaptation.