# Research

## Assignment Flows: Dynamical Systems on Riemannian Manifolds for Data Analysis

This project is funded by the DFG within the Priority Programme on the Theoretical Foundations of Deep Learning

**Scope.** We study product spaces of elementary Riemannian manifolds for the context-sensitive analysis of data observed in any metric space. State spaces interact dynamically by geometric averaging and locally according to the adjacency structure of an underlying graph. The corresponding interaction parameters are learned from data. Geometric integration of the resulting continuous-time flow generates layers of a neural network. Our approach enables to study dynamical relations of inference and learning in neural networks from a geometric viewpoint, along with a probabilistic interpretation of contextual decision making. From the numerical point of view, the approach copes with high dimensions and large problem sizes.

**Mathematical aspects.** Information geometry, coupled and regularized information transport, geometric mechanics on manifolds and variational principles, geometric numerical integration, statistical performance characterization using PAC-Bayesian analysis.

**Recent work.** download

- confluence of assignment flows and Laplace-Beltrami flows characterising data labeling as generalized harmonic maps (arXiv:2408.15946, 2024)
- extension to generative assignment flows for discrete joint distributions (arXiv:2402.07846, 2024)
- geometric embedding approach to multiple games and populations (arXiv::2401.05918, 2024)
- quantum state assignment flows (Entropy, 2023)
- geometric mechanics of assignment flows (Information Geometry, 2023)
- novel PAC-Bayes bound for structured prediction (NeurIPS, 2023)
- self-certifying classification by linearized deep assignment flows (PAMM, 2023)
- non-local graph PDE for structured labeling based on assignment flows (SIAM SIIMS, 2023)
- learning linearized assignment flows (JMIV, 2023)
- convergence and stability of assignment flows (Information Geometry, 2022)
- continuous-domain assignment flows (Europ. J. Appl. Math., 2021)
- order-constrained 3D OCT segmentation using assignment flows (IJCV, 2021)
- self-assignment flows (SIAM SIIMS, 2020)
- unsupervised assignment flows (JMIV, 2020)
- geometric integration of assignment flows (Inverse Problems, 2020)
- assignment flows for labeling: introduction (Handbook: Var. Meth. Nonl. Geom. Data, 2020)
- assignment flows for metric data labeling (JMIV, 2017)