Organizers: Carsten Hartmann, Sebastian Reich

Abstract:

Generative diffusion models can transform a data distribution to noise and remove the noise by a reverse transformation to obtain new data with a distribution similar to the original one; they offer state of the art performance in generative AI for images. Normalizing Flows are deterministic or stochastic generative models that  produce tractable distributions where both sampling and parameter estimation can be exact or efficient; they have strong connections to data assimilation and optimal control. The goal of this minisymposium is to survey recent progress in the aforementioned fields of research and to discuss connections between them. A particular focus is on the application to multiscale problems and stochastic control.

Speakers:

14:30h: Andrew Duncan (Imperial College London)

Diffusion Paths for Sampling: From Stochastic Averaging to Sequential Monte Carlo

Sampling from high-dimensional, non-log-concave distributions is a central challenge in statistics, Bayesian inference, and generative modelling. Classical annealing methods address this problem by introducing a sequence of intermediate distributions that gradually bridge a simple reference measure and a complex target. While effective in principle, their performance can be highly sensitive to discretisation choices, path geometry, and the quality of gradient information available along the annealing schedule. We present an alternative construction inspired by score-based diffusion models, in which the target is connected to a simple reference distribution through a continuous diffusion path. Sampling can then be formulated in terms of stochastic dynamics evolving along this path. A central computational difficulty is that the score of the intermediate distributions is typically unavailable, even when the target density itself can be evaluated. We discuss two complementary strategies that make diffusion-path sampling practical, crucially, without requiring any training of neural score models, and discuss non-asymptotic convergence guarantees that show how properties of the diffusion path govern finite-time sampling error.

15:00h: Manfred Opper (TU Berlin)

Generative diffusion models with fractional Brownian noise

Generative diffusion models aim at constructing a stochastic process which bridges two probability densities (implicitly specified by large data samples) as marginals at initial and final times. If the first, the source density, is easy to sample from, one can generate e.g. data from the second, the target density, by sampling from the trained model.

While much work so far has been based on processes defined by white noise driven stochastic differential equations, we will discuss in this talk a generalisation to a class of coloured noise processes, the so-called fractional Brownian motion. While these processes are non-Markovian, we develop approximations which allow us to efficiently train the
corresponding stochastic bridge models. We will show applications to generation of data from noise and of paired data translations.

15:30h: Claudia Strauch (Heidelberg University)

Diffusion modelling beyond Ornstein–Uhlenbeck: statistical analysis of random stopping and reflections

We discuss two approaches to modifying the dynamics of generative diffusion models, tailoring them to specific geometric or topological constraints of the target distribution. First, we briefly outline Denoising Reflected Diffusion Models, which introduce reflection terms to constrain the process to a bounded domain. We then focus on a class of diffusion models achieving a time-homogeneous structure via Doob's h-transform, which terminates the process at a suitable sampling distribution at a random time. The model is particularly well suited to generating data with lower intrinsic dimensions, as the termination criterion simplifies to a first-hitting rule. Without relying on the explicit analytical structure of standard Ornstein–Uhlenbeck processes, we develop alternative analytical tools to establish statistical guarantees depending on the intrinsic geometry of the data.