Barron Optimal Transport I: Generative Modeling

2026-10-07

AI Summary

This study addresses the inefficiency of neural representations in generative models and the lack of quantitative theory characterizing the suboptimality of diffusion models. To this end, it introduces a Barron optimal transport framework that penalizes neural network complexity. By substituting the conventional mean kinetic energy with the Benamou–Brenier dynamical formulation under the Barron norm, a novel metric space is constructed to analyze the geometric properties of generative models. This work elucidates the adaptive mechanisms underlying feature learning and establishes super-polynomial lower bounds for score approximation. Furthermore, it quantifies the suboptimality of diffusion models within the proposed Barron geometry and extends the applicability of Stein variational gradient descent.

Abstract

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 $L^2$ energy by the \emph{Barron} energy \cite{bach2017breaking, 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 \cite{companionpaper} we leverage the Barron transport geometry for sampling applications, extending the scope of Stein variational gradient methods via feature adaptation.

Problem

• Optimal Transport
• Generative Modeling
• Neural Network Complexity
• Barron Energy
• Diffusion Models

Innovation

• Optimal Transport
• Generative Modeling
• Barron Energy
• Neural Network Complexity
• Diffusion Models