The Fundamental Limits of Valid Transport Map Estimation
2026-06-29 • Machine Learning
Machine Learning
AI summaryⓘ
The authors look at how some modern generative models, like diffusion models and flow matching, learn ways to move data from one form to another without focusing on the best possible way (optimal transport). They create a clear framework to study how hard it is to learn any valid transport method, showing that under common assumptions, learning any valid transport is about as hard as learning the optimal one. However, if these assumptions don’t hold, learning easier-to-find transport maps can be more accurate. Their work helps explain when it’s useful to settle for less-than-optimal transport maps in generative modeling.
generative modelingdiffusion modelsnormalizing flowsflow matchingoptimal transporttransport mapsminimax frameworksample complexitystatistical stabilitytransport plan
Authors
Sivaraman Balakrishnan
Abstract
Many modern generative modeling methods, including diffusion models, normalizing flows, and flow matching, estimate transport maps or plans between distributions without explicitly targeting an optimal transport (OT) map. In applications like generative modeling, the transport cost itself is irrelevant, and this makes it natural to target maps which are more tractable from either a statistical or computational standpoint. In this short note, we formalize the task of estimating any valid transport map in a rigorous minimax framework. One consequence of this framing is that it yields sample complexity lower bounds for any method whose learned object is evaluated as a transport map or plan, including flow matching and diffusion-based generative models, in settings where direct analysis would be challenging due to the analytic complexity of the methods and their target maps. We observe that, under standard, though strong, stability assumptions from the OT literature, estimating any valid transport map is statistically as hard as estimating the OT map. We complement these results with some examples showing that when these stability assumptions fail, alternative transport maps can be learned substantially more accurately than the OT map. Our minimax framing provides a rigorous foundation for understanding the statistical limits of modern transport-based generative methods and clarifies when targeting sub-optimal maps can provide real statistical advantages.