On approximation and estimation of Schrödinger potentials without the curse of dimensionality

We examine generative modelling approaches based on the construction of Schrödinger bridges between Gaussian noise and a target distribution. It is known that the solution of the dynamic Schrödinger problem is a diffusion process with a drift associated with Doob's h-transform of a Schrödinger potential. Although its accurate restoration from finite samples is crucial for reliable, high-quality data generation, the existing literature lacks theoretical guarantees regarding this question. In our work, we establish theoretical upper bounds on the complexity of Schrödinger potential approximation and estimation via neural networks. These bounds are determined by the effective dimension of the target distribution. To our knowledge, this is the first result demonstrating that generative modelling methods based on Schrödinger bridges and stochastic optimal control can escape the curse of dimensionality.