Convexity and Grassmannians in Statistical Inference

• Applicant: Professor Dr. Martin Wahl
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539663130

Future progress in statistics and data science will in many cases depend on combinatorial, algebraic and geometric insights. While the ambient spaces for data representations tend to be high-dimensional, data actually encountered in applications is sampled from low-dimensional structures. These are often given by differentiable manifolds, algebraic varieties, or shape constraints. Fruitful mathematical approaches to such structures include topological data analysis, information geometry, and algebraic statistics. These fields are foundational for novel algorithms in statistics and data science, in contexts as diverse as manifold learning, non-parametric statistics, maximum likelihood estimation, and Bayesian inference. The goal of this project is to advance our understanding of statistical models that involve a significant combinatorial and geometric component. We investigate uniform distributions on polytopes and convex bodies, covariance matrices contained in certain linear spaces of symmetric matrices, and discrete statistical models described by Grassmannians. We adopt an exact and non-asymptotic point of view, with small to moderate number of observations. This allows us to combine methods from statistics and computer algebra, in order to achieve mathematically optimal and precise solutions.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies
• Co-Investigator: Professor Bernd Sturmfels, Ph.D.