Superlinear drift models lower transport costs beyond Ornstein Uhlenbeck
Ornstein-Uhlenbeck Is Hard to Beat, Yet Superlinear Drift Ships Lower Transport Costs
Machine Learning
Summary
The Ornstein-Uhlenbeck process is a popular mathematical model used for simulating how things evolve over time. Previous work showed it was very hard to outperform when the model's drift doesn't grow faster than linear. This paper explores models with drift that grows faster than linear and finds these can do better in tasks like generating images, showing more consistent results and lower costs. This suggests the previous belief that Ornstein-Uhlenbeck is the best is not always true.
What this means in practice
- •For machine learning engineers: Improve image generation models by using superlinear drift Langevin diffusions for better sample quality and stability.
- •For computational physicists: Simulate complex particle transport phenomena more efficiently using superlinear drift models for lower transport costs.
Authors
Attila Lovas, Lóránt Nagy
Abstract
Brešar and Mijatović \cite{bresar2025} show that Ornstein--Uhlenbeck diffusion is hard to beat in forward convergence under assumptions that exclude superlinear drift. We instead test superlinear Langevin diffusions for score-based image generation, computing their conditional scores numerically from a Fokker--Planck equation. In our experiments, the superlinear models beat the Ornstein--Uhlenbeck baseline on empirical Wasserstein distance across nearly the entire tested grid and show less variation across diffusion horizons. The ``hard to beat'' verdict of \cite{bresar2025} thus fails to be universal.