Research Group: Statistics of Stochastic Processes

Over the last decade, generative modelling has emerged as a powerful probabilistic tool in the field of machine learning. The concept is straightforward: transform noise to generate new data that aligns with a specific training dataset. These transformations must adapt to the information contained in the training data, which is typically high-dimensional. One class of generative model that has demonstrated highly proficient empirical generation capabilities in various contexts are denoising diffusion models.

To explain their empirical success mathematically, we must identify scenarios in which the distance between the target and generated distributions converges at a sufficiently fast rate (ideally minimax optimal) in terms of sample size, intrinsic dimension, and data distribution smoothness. Although significant progress has been made on related statistical questions in more idealised settings, existing statistical theory still falls short of providing a convincing mathematical explanation for why denoising diffusion models perform so well across a wide range of tasks. Our work focuses on understanding the statistical properties of denoising diffusion models in non-standard settings.

Contact: Asbjørn Holk Thomsen

Although stochastic control problems are fairly well understood for many relevant systems, solutions provided by appropriate control strategies usually require specific knowledge of the system and its dynamics. In practice, this is often only partially feasible in that, while the system structure may be justified - for example, by modelling the dynamics via a stochastic differential equation (SDE) - the exact components are not available. The joint problem of estimating system components, or system identification, and simultaneous control then arises.

We are particularly interested in the corresponding nonparametric statistical challenges of estimating infinite-dimensional objects, and in developing control strategies based on these estimators for SDE-type systems. Our main focus is on developing and evaluating different learning strategies for stochastic control that utilise a close link to the theory of stopping times in stochastic processes.

Contact: Hans Reimann

Stochastic partial differential equations (SPDEs) are used to describe the evolution of dynamical systems in the presence of persistent spatial-temporal uncertainties. In practice, the irregularities of many complex systems in various contexts are often too detailed for precise mathematical description. At this point, it is useful to introduce noise as a phenomenological replacement for a realistic element discarded by the deterministic part of the mathematical model.

Both the analytical and probabilistic theory of SPDEs and the statistical inference thereof have recently seen significant development, paving the way for the realistic modelling of complex phenomena in many applied disciplines, such as biology, finance, fluid dynamics, geostochastics, neuroscience and population growth. While the general form of a particular SPDE is commonly derived from the fundamental properties of the underlying processes under study, parameters often arise in the formulation that need to be specified or determined based on empirical observations. In such situations, the problem of parameter estimation emerges naturally. Assuming that a phenomenon of interest follows the dynamics of an SPDE and that one or more realisations of this process have been measured, we aim to determine the unknown parameters in the model to ensure that the equations fit or predict the observed data as accurately as possible.

As well as applying and investigating familiar estimation techniques in the context of relevant SPDE models, we are also interested in broadening the scope of statistical methodology by developing new parameter estimation strategies.

Contact: Timo Dörzbach