### PhD Students

**Max Engel** (PhD project: Compiling Einsum)

**Markus Fischer** (PhD project: Tensor Networks for Classical and Quantum Codes)

Classical codes, such as Turbo Codes, are given as graphical models that can be represented as tensor networks. Tensor network contraction can then be used for inference. Quantum codes are often naturally given as tensor networks. In this PhD project we will explore tensor contraction algorithms for fast inference in classical and quantum codes.
**Jaroslav Garvardt** (PhD project: Algorithms for Hard Clustering Problems and Bayesian Network Structure Learning)

Many optimization problems in data science are NP-hard. Jaroslav Garvardt investigates whether one can develop efficient algorithms for generally hard problems by exploiting the structure of type-safe inputs. His current focus is on the modularity clustering problem, one of the most important graph-based clustering methods. Jaroslav Garvardt is also researching efficient algorithms for Bayesian network structure learning, where a graphical model is to be learned from observed data.
**Tim Hoffmann** (PhD project: Proof Systems for Model Counting)

Model counting is an extension of SAT solving to the more complex problem of determining the number of satisfying solutions (#SAT). This problem has many applications, in particular in probabilistic reasoning. In the project we evaluate the strength of current reasoning methods for model counting and formalise and investigate proofs for #SAT solvers.
**Kaspar Kasche** (PhD project: Proof Complexity)

Proof complexity of Quantified Boolean Formulas (QBF) investigates the size of proofs of QBFs in central proof proof systems, including various resolution calculi. Lower bound methods for these systems have deep connections to logic and circuit complexity. A particular focus of this project will be to investigate the relations to communication complexity. QBF proof complexity provides the main theoretical tool to assess the strength of modern and versatile QBF solvers and aims to guide their future developments.
**Andreas Kröpelin** (associated PhD project: Statistical Methods for Reconstruction of Continuous Movements from Cryo-EM Data)

Cryo-electron microscopy (EM) does not only determine biomolecular structures at high resolution, but shows also potential for unraveling the entire spectrum of conformational dynamics of macromolecular assemblies. Our project aims to develop new computational approaches to reconstruct deformable 3D models that capture all of the structural dynamics hidden in cryo-EM projection images. The additional information about the dynamics of proteins will lead to a deeper understanding of enzymatic mechanisms and provide new insights into medically relevant modes of inhibiting proteins by drugs and binders.
**Farin Lippmann** (PhD project: Nested Sampling with log-Barriers)

Nested sampling is a powerful approximate algorithm for Bayesian inference that allows the generation of posterior samples and the evaluation of the model evidence. A crucial and challenging step is to generate samples from the prior restricted by a lower bound on the likelihood function. In this project, we explore the use of methods from constrained optimization such as log-barrier functions in order to improve the efficiency of sampling from the restricted prior.
**Niklas Merk** (PhD project: Reinforcement Learning of Tensor Contraction Orders)

For evaluating tensor expressions, the key task is to determine the order in which to contract the tensors. Unfortunately, finding an optimal tensor contraction order is a NP-hard problem. Therefore, one has to resort to heuristic algorithms, for instance, greedy heuristics, for computing contraction orders. In this PhD project an alternative approach is pursued, namely, using reinforcement learning for learning good contraction orders.
**Julian Möbius** (PhD project: Modeling and Sampling of Biomolecular Structures with Stochastic Interpolants)

Diffusion models, normalizing flows, as well as energy- and score-based models are members of a versatile family of probabilistic models that show impressive performance in probabilistic modeling and sampling. These models allow us to bridge between two or more distributions and have been termed stochastic interpolants. In this project, I explore the use of stochastic interpolants in posterior sampling and applications of Bayesian inference to modeling biomolecular structure and dynamics.
**Jurek Rostalsky** (PhD project: Combinatorial Scientific Computing)

Combinatorial problems on graphs are often considered in isolation from numerical problems in mathematical optimisation. The goal of this project is to explore synergies between discrete mathematics, theoretical computer science, and scientific computing.
**Paul Gerhardt Rump** (PhD project: Efficient Tensor Calculus and Convexity Certificates)

Matrix calculus computes derivatives of linear algebra expressions in vectorized form. The derivative of such an expression with respect to matrix is a matrix expression and its Hessian is a fourth order tensor, which cannot be expressed in a vectorized linear algebra language. For analyzing Hessians of matrix dependent functions it is necessary to work with a more general language for tensor expressions. The project aims at computing succinct and easy to analyze expressions for derivatives of tensor dependent expressions. The expressions can be used for computing convexity certificates for tensor dependent expressions.
**Philip Schär** (associated PhD project: Slice Sampling: Theoretical and Methodological Advances)

Sampling from intractable distributions over high dimensional Euclidean spaces is a central problem in Bayesian inference. One of the standard approaches to generate approximate solutions to this problem is Markov chain Monte Carlo (MCMC). We investigate a certain class of MCMC methods known as slice sampling, with a particular focus on polar slice sampling (Roberts & Rosenthal, 2002) and variations thereof. On the one hand, we aim to improve the theoretical understanding of slice sampling, e.g. by proving broader and/or sharper convergence rate estimates. On the other hand, we aim to develop new sampling methods, e.g. by modifying an existing algorithm to enable efficient implementation, and carefully investigate their empirical performance.
**Luca Staus** (PhD project: Algorithm Engineering and Multivariate Algorithmics for Learning and Reoptimization of Optimal Inference Engines)

**Henrik Voigt** (PhD project: From Token Prediction to Rule Prediction)

The transformer model has revolutionzied natural language processing and more. A its core, the tranformer model is a probabilistic model for predicting the next language token from a prefix of tokens. This PhD project will explore transformer models that, instead of tokens, predict the rule from a formal grammar that should be used to derive the next symbol.
**Maurice Wenig** (PhD project: Tensor Network Language Models)

Like many physical systems, natural language exhibits long range correlations. In (quantum) statistical physics, long range correlations have been modeled succesfully by tensor networks. It has been suggested to use these models also for natural language. In this PhD project, we will pursue this idea further.
**Brian Zahoransky** (PhD project: Visualization and Exploration of Cryo-EM Data and Gaussian Mixture Models)

This project is mainly concerned with probabilistic machine learning and modeling, computational Bayesian statistics and the analysis of complex biological data. Our goal is to visualize Cyro-EM data and to develop interactive exploration tools. The project is a cooperation between the Visualization group under Kai Lawonn and the Microscopic Image Analysis group at the University Hospital Jena under Michael Habeck. The working group is mainly concerned with probabilistic machine learning and modeling, computational Bayesian statistics, and the analysis of complex biological data. Our goal is to visualize the Cyro-EM data provided by the research group and to develop interactive exploration tools.
### Faculty

**Olaf Beyersdorff** (computational logic)

**Torsten Bosse** (automatic differentiation)

**Alexander Breuer** (high-performance computing)

**Martin Bücker** (automatic differentiation and high-performance computing)

**Wolfhart Feldmeier** (scientific machine learning)

**Joachim Giesen** (artificial intelligence and algorithm engineering)

**Michael Habeck** (probabilistic inference and bio-physics)

**Christian Komusiewicz** (algorithm engineering and artificial intelligence)

**Kai Lawonn** (visualization)

### Coordination

**Olia Blacher** (project coordinator)

**Silvia Blaser** (administrative assistance)

### Advisory Board

**Anthony G Cohn** (University of Leeds, UK)

**Thomas Eiter** (TU Wien, AT)

**Vijay Ganesh** (University of Waterloo, CA)

**Angelika Kimmig** (Katholieke Universiteit Leuven, BE)

**Meena Mahajan** (The Institute of Mathematical Sciences (IMSc) Chennai, IN)

**Tom Minka** (Microsoft Research, UK)

**Peter Schuster** (Universita degli Studi di Verona, IT)