Marginal Independence Models
Tobias Boege (MPI MiS)
We impose rank one constraints on marginalizations of a tensor given by a simplicial complex. If the tensor encodes a discrete probability distribution, the rank constraints correspond to mutual independences among the random variables. These statistical models become toric after a linear change of coordinates. We study their toric ideals with emphasis on random graph models and use the new coordinates to compute degrees for their parameter estimation problems.
Signal processing on graphs and complexes
Michael Schaub (RWTH Aachen University)
Graph signal processing (GSP) tries to device appropriate tools to process signals supported on graphs by generalizing classical methods from signal processing of time-series and images -- such as smoothing, filtering and interpolation -- to signals supported on the nodes of a graph. Typically, this involves leveraging the structure of the graph as encoded in the spectral properties of the graph Laplacian. In certain scenarios, such as traffic network analysis, the signals of interest are however naturally defined as flows the edges of a graph, rather than on the nodes. After a brief recap of the central ideas of GSP, we examine why standard tools from GSP may not be suitable for the analysis of such flow signals. More specifically, we discuss how the underlying notion of 'signal vs noise' inherited from typically considered variants of the graph Laplacian are not suitable when dealing with edge signals that encode flows. To overcome this limitation, we devise signal processing tools based on the Hodge-Laplacian and the associated discrete Hodge Theory for simplicial (and cellular) complexes. We discuss applications of these ideas for signal smoothing, semi-supervised and active learning for edge-flows on discrete or discretized spaces.
On eigenvalues of symmetric matrices with PSD principal submatrices
Khazhgali Kozhasov (TU Braunschweig)
Real symmetric matrices of size n, whose all principal submatrices of size k are positive semi-definite, form a closed convex cone. Such matrices do not need to be PSD and, in particular, they can have negative eigenvalues. The geometry of the set of eigenvalues of all such matrices is far from being completely understood. In this talk I will show that, already when (n,k)=(4,2), the set of eigenvalues is not convex.
Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture
Jonathan Leake (Weierstrass Institute)
About 5 years ago, the Heron-Rota-Welsh conjecture (log-concavity of the coefficients of the characteristic polynomial of a matroid) was proven by Adiprasito, Huh, and Katz via the exciting development of a new combinatorial Hodge theory for matroids. In recent work with Petter Brändén, we have given a new short "polynomial proof" of the Heron-Rota-Welsh conjecture. Our proof uses an extension of the theory of Lorentzian polynomials to convex cones, which generalizes real stable and hyperbolic polynomials. In this talk, I will briefly discuss the basics of Lorentzian (aka completely log-concave) polynomials, and then I will give an overview of our new proof of the Heron-Rota-Welsh conjecture.
AAA in-person meeting
During the AAA in-person meeting representatives of local groups from Braunschweig and Osnabrück present their research results. The event is aiming at fostering collaborations between the two universities.
The morning session (10:20 - 12:40) takes place in the lecture hall SN 23.1, the afternoon session (14:00 - 15:30) will be in the lecture hall PK 4.1. We plan to stream all talks at the usual link.
| 10:20 - 10:50 | Learning variational models with unrolling and bilevel optimization | Niklas Breustedt (TU Braunschweig) |
| 10:50 - 11:20 | A nonlinear randomized Kaczmarz method with Bregman projections | Maximilian Winkler (TU Braunschweig) |
| 11:20 - 11:40 | Break | |
| 11:40 - 12:10 | Graph and distributed extensions of the Douglas-Rachford method | Emanuele Naldi (TU Braunschweig) |
| 12:10 - 12:40 | Object Detection for nanoscale images | Mandy Stritzke (TU Braunschweig) |
| 12:40 - 14:00 | Lunch | |
| 14:00 - 14:30 | Sparse signals on graphs | Tarek Emmrich (Universität Osnabrück) |
| 14:30 - 15:00 | Gordan's lemma up to symmetry | Dinh Le Van (Universität Osnabrück) |
| 15:00 - 15:30 | On polyhedral invariants of polynomial maps on the plane | Boulos El Hilany (TU Braunschweig) |
Nonarchimedean Integral Geometry
Antonio Lerario (SISSA)
In this seminar I will explain a recent construction, introduced in a joint work with P. Bürgisser and A. Kulkarni, for studying the "Riemannian geometry" of nonarchimedean manifolds. This construction uses a nonarchimedean version of Sard's lemma and the coarea formula, and leads to a nonarchimedean version of integral geometry, with applications to Schubert calculus and fewnomial systems.