Deep learning for inverse imaging problems: some recent approaches
In this talk we discuss the idea of data-driven regularisers for inverse imaging problems. We are in particular interested in the combination of model-based and purely data-driven image processing approaches. In this context we will make a journey from "shallow" learning for computing optimal parameters for variational regularisation models by bilevel optimization to the investigation of different approaches that use deep neural networks for solving inverse imaging problems. Alongside all approaches that are being discussed, their numerical solution and available solution guarantees will be stated.
Inference on mechanistic network models and on spreading processes evolving on networks
Many systems of scientific interest can be investigated as networks, where nodes correspond to the elements of the system and edges to interactions between the elements. Increasing availability of large-scale data and steady improvements in computational capacity are continuing to fuel the growth of this field. Network models are now commonly used to investigate social, economical and biological complexity at the systemic level. There is a general divide between the two prominent paradigms to the modeling of networks, which are the approach of mechanistic networks models and the approach of statistical network models. Mechanistic models are knowledge domain driven and assume that the microscopic mechanisms governing network formation and evolution at the level of individual nodes are known, and questions often focus on understanding macroscopic features that emerge from repeated application of these known mechanisms. The statistical approach, in contrast, often starts from observed network structures and attempts to infer some aspects about the underlying data generating process. Mechanistic network models provide insight into how the network is formed and how it evolves at the level of individual nodes, but as mechanistic rules typically lead to complex network structures, it is difficult to assign a probability to any given network realizations that a mechanistic model may generate. Because of this difficulty, there is typically no closed form expression for the likelihood for these models and, consequently, both likelihood and posterior based inference for learning from data is not possible. We have developed a principled statistical framework, based on Approximate Bayesian Computation, to bring some of the mechanistic network models into the realm of statistical inference both for parameter estimation, construction of confidence/credible intervals, hypothesis testing and model selection. I will introduce the general Approximate Bayesian Computation (ABC) framework and demonstrate its application to mechanistic networks, where it can be used to infer model parameters, and their associated uncertainties, from empirical data. In the second part of the talk I will move to inference on stochastic processes that evolve over a given network structure. In particular I will focus on infectious diseases to understand their spreading mechanisms, to evaluate control strategies and to predict the risk and course of future outbreaks. Although the underlying processes of transmission are different, the network approach can be used to study the spread of pathogens in a contact network or the spread of rumors in an online social network. We study simulated simple and complex epidemics on synthetic networks and on two empirical networks, a social / contact network in an Indian village and an online social network in the U.S. Our goal is to learn simultaneously about the spreading process parameters and the source node (first infected node) of the epidemic, given observations about state of nodes at several points in time. Also in this context ABC enables us to adopt a Bayesian approach to the problem despite the posterior distribution being very complex.
Joint work with Jukka-Pekka Onnela (Department of Biostatistics, Harvard University, US) and Ritabrata Dutta (Warwick University)