Deep Learning meets statistics: Circumventing the curse of dimensionality with deep neural networksRoom B052 (Haus B , 80333 Theresienstr. 39)

A combination of algorithmic advances together with computational power and availability of data have led to a remarkable empirical success of deep learning across a range of domains. The effectiveness of the neural networks stands in stark contrast to the theoretical understanding of the procedure. Currently, we are far from a mature, unified theoretical theory of deep learning. Nevertheless, interesting results exist, and one can roughly divide the existing literature in three parts, namely approximation, generalization and optimization. In this talk we mainly focus on generalization results. In particular, we contribute to the question why neural networks perform well on unknown new data. From a statistical learning perspective, one might answer this question by analyzing neural networks as estimators in a nonparametric regression setting. In recent results it was shown that neural networks are able to circumvent the so?called curse of dimensionality in case that suitable restrictions on the structure of the regression function hold. Under a general composition assumption on the regression function, one key feature of the neural networks used in these results is that their network architecture has a further constraint, namely the network sparsity. In this talk we show that we can get similar results also for least squares estimates based on simple fully connected neural networks with ReLU activation functions. Our analysis is twofold. In a first result, we show that fully connected networks derive a rate of convergence independent of the input dimension in case that the regression function fulfills some hierarchical composition structure. In a second step, the case, that the distribution of the predictor variable is concentrated on a manifold, is analyzed. As an outlook of this talk, we present finally a result which analyzes all three aspects of deep learning, namely approximation, generalization and optimization, simultaneously. In a simplified setting, i.e., for neural networks with one hidden layer, we show that neural networks learned by gradient descent circumvent the curse of dimensionality in case that the Fourier transform of the regression functions decays suitable fast.

Equation-based, object-oriented modelling and its application for the simulation of complex thermal architectures Online: attend (Code 101816)MI 03.04.011 (Boltzmannstr. 3, 85748 Garching)

Equation-based modelling languages such as Modelica enable the description of components and subsystems in mathematical terms. In industrial practice this is used to organize the knowledge in dedicated libraries and to design and control complex dynamical systems. This presentation shows the fundamental building blocks of such declarative computer languages and demonstrates the domain-specific challenges that need to be overcome for the simulation of modern thermal architectures as they appear in electric cars or environmental control systems of aircraft.

Link and Passcode: https://tum-conf.zoom.us/j/96536097137 Code 101816

Rigorous mean-field derivation of cross-diffusion population modelsvia ZOOM (Boltzmannstr. 3, 85748 Garching)

In the field of population dynamics, cross-diffusion partial differential equations have gained more impact, see for instance the SKT-model by Shigesada, Kawasaki and Teramoto. However, a rigorous derivation starting from a stochastic many-particle system was still missing in the literature. In this talk, I will show how the approach of moderately interacting particles in the mean-field limit can be used in order to derive cross-diffusion models of SKT-type starting from a stochastic interacting many-particle system. The talk will also include a short introduction to mean-field limits, their connection to PDEs and the concept of moderately interacting particles. As a byproduct of the mean-field derivation, we also study a non-local version of the underlying PDE model which represents an intermediate level between the particle dynamics and the cross-diffusion partial differential equation. This talk is based on the joint work with Li Chen, Esther Daus and Ansgar Jüngel ' Rigorous derivation of population cross-diffusion systems from moderately interacting particle systems`, Journal of Nonlinear Science, 31(6), 1-38 (2021).

Voting-based probabilistic consensuses and their applications in distributed ledgersOnline: attendB 252 (Theresienstraße 39, 80333 München)

First, we give a short introduction in probabilistic consensus protocols and their applications in distributed ledgers. Then, we discuss the problem of metastability and how it can be resolved using common random numbers. After that, we present a current proposal, the so called On Tangle Voting (OTV), that uses the ledger itself as a voting layer. Finally, we conclude with first theoretical and numerical results, and several open questions. Joint work with the IOTA Foundation.

The extremal graphical lassoOnline: attendBC1 2.01.10 (Parkring 11, 85748 Garching)

Multivariate extreme value theory is interested in the dependence structure of multivariate data in the unobserved far tail regions. Multiple characterizations and models exist for such extremal dependence structure. However, statistical inference for those extremal dependence models uses merely a fraction of the available data, which drastically reduces the effective sample size, creating challenges even in moderate dimension. Engelke & Hitz (2020, JRSSB) introduced graphical modelling for multivariate extremes, allowing for enforced sparsity in moderate- to high-dimensional settings. Yet, the model selection and estimation tools that appear therein are limited to simple graph structures. In this work, we propose a novel, scalable method for selection and estimation of extremal graphical models that makes no assumption on the underlying graph structure. It is based on existing tools for Gaussian graphical model selection such as the graphical lasso and the neighborhood selection approach of Meinshausen & Bühlmann (2006, Ann. Stat.). Model selection consistency is established in sparse regimes where the dimension is allowed to be exponentially larger than the effective sample size. Preliminary simulation results appear to support the theoretical results.

Feynman-Kac particle models for cloning algorithmsOnline: attend (Code 101816)MI 03.04.011 (Boltzmannstr. 3, 85748 Garching)

Fluctuations of dynamic observables in probabilistic models of physical systems, such as particle or energy currents, can be studied in the framework of large deviations. Probabilities of rare events typically decay exponentially with measurement time, and an explicit derivation of the decay rate (the rate function) is often not possible. To this end, efficient numerical algorithms have been developed in the statistical physics literature, based on importance sampling via cloning of rare event trajectories. Adapting previous results from sequential Monte Carlo methods, we use Feynman-Kac models to establish fully rigorous bounds on systematic and random errors of such cloning algorithms in continuous time. We develop a method to compare different algorithms for particular classes of observables, based on the martingale characterization of stochastic processes. Our results apply to a large class of jump processes on compact state space, and provide a framework that can also be used to evaluate and improve the efficiency of algorithms. This is joint work with Letizia Angeli, Adam Johansen and Andrea Pizzoferrato.

Link and Passcode: https://tum-conf.zoom.us/j/96536097137 Code 101816

Safe spaces in geometry and analysis Online: attend

Abstract: In the Wild West of geometry and groups, some familiar objects feel like home: Euclidean spaces, hyperbolic geometry, and more generally all the geometries described by semisimple Lie groups or similar matrix groups over local fields. Harmonic analysis and operator algebras have their own wild oceans, but again with a small safe haven: the "Type I", home of the commutative world, compact groups, generalizations of Fourier analysis. In a precise sense, objects that can be "described". I will present a connection between these two worlds and show how it leads to previously unexpected classification results.

Hybrid-Veranstaltung: Präsenzvortrag im Hörsaal A027 und gleichzeitige Übertragung per Zoom mit folgendem Link:

Join Zoom Meeting: https://lmu-munich.zoom.us/j/99946902916?pwd=UWM5SGtIL091NmdjU3BHVVpOU0lEdz09

Meeting ID: 999 4690 2916, Passcode: 695211

Poisson and non-Poisson Chimeras in Two-population Networksvia ZOOM (Boltzmannstr. 3, 85748 Garching)

Chimera states, i.e. dynamical states composed of two apparently incongruent parts in an ensemble of coupled identical oscillators, namely synchronous and asynchronous oscillations, have been proven to be important symmetry-broken dynamical states in various topologies of oscillators in both natural and man-made systems. From a mathematical view point, a finite-sized network consisting of two subpopulations is in particular suitable to study chimera states since this topological structure possesses inherently an invariant synchronized manifold, in contrast to a system of spatially distributed oscillators along a one-dimensional ring.

In this talk, we study dynamical and spectral properties of finite-sized chimeras on two-population networks. We investigate the Kuramoto order parameter dynamics based on the initial condition and the system size dependence. We elucidate how the Kuramoto order parameter of the finite sized globally coupled two-population network of phase oscillators is connected to that of the continuum limit. Based on the dependence of the observable chimera dynamics we define a Poisson chimera and a non-Poisson chimera. We perform a Lyapunov analysis of these two types of chimera states and elucidate the spectral properties of them using both the reduced macroscopic dynamics (Watanabe-Strogatz or Ott-Antonsen theory) and the network symmetry-induced cluster pattern dynamics. The numerically performed Lyapunov analysis suggests that the chimera states are neutrally stable except for the macroscopic dynamics. Next, we consider a nonlocal intra-population network and a network of Stuart-Landau oscillators that possess amplitude degrees of freedom. We demonstrate that both variations may act as small heterogeneities that render Poisson chimeras attracting.

Title: Statistical topological data analysisOnline: attend

We study selected statistical and probabilistic topics in persistent homology, which is the major branch of topological data analysis (TDA). TDA itself refers to a collection of statistical methods that find topological structure in data. Persistent homology is a multiscale approach to quantifying topological features in data, in particular, point cloud data.After a short and heuristic introduction to the main ideas of persistent homology, we will study multivariate and functional central limit theorems (CLT) and related stabilizing properties for persistent Betti numbers in the critical regime. Based on the multivariate CLT, we consider a smooth bootstrap procedure to construct confidence intervals. More- over, using a functional central limit theorem, we derive goodness-of-fit test in order to compare network structures for different types of underlying point processes.

Estimation and Inference for Differential NetworksOnline: attendBC1 2.01.10 (Parkring 11, 85748 Garching)

We present a recent line of work on estimating differential networks and conducting statistical inference about parameters in a high-dimensional setting. First, we consider a Gaussian setting and show how to directly learn the difference between the graph structures. A debiasing procedure will be presented for construction of an asymptotically normal estimator of the difference. Next, building on the first part, we show how to learn the difference between two graphical models with latent variables. Linear convergence rate is established for an alternating gradient descent procedure with correct initialization. Simulation studies illustrate performance of the procedure. We also illustrate the procedure on an application in neuroscience. Finally, we will discuss how to do statistical inference on the differential networks when data are not Gaussian.

Transient dynamics in stochastic systemsvia ZOOM (Boltzmannstr. 3, 85748 Garching)

Transient dynamics, often observed in multi-scale systems, are roughly defined to be the interesting dynamical behaviours that display over finite time periods. For a class of randomly perturbed dynamical systems that arise in chemical reactions and population dynamics, and that exhibit persistence dynamics over finite time periods and extinction dynamics in the long run, we use quasi-stationary distributions (QSDs) to rigorously capture the transient states governing the long transient dynamics. We study the noise-vanishing concentration of the QSDs to gain information about the transient states and investigate the dynamics near transient states to understand the transient dynamical behaviours as well as the global multiscale dynamics. We expect that the QSD framework provides a useful way to studying transient dynamics of different mechanisms.

A functional version of Kesten and Stigum's theoremB 252 (Theresienstr. 39, 80333 München)

If $(Z_n(\lambda))_{n\geq 0}$ is a discrete time Galton-Watson process with mean offspring $\lambda$, then $Z_n(\lambda)/\lambda^n$ is a non-negative martingale which converges almost surely to a limit $W(\lambda)$. In this joint work with JF Marckert, we define a process of Galton-Watson processes indexed by the mean offspring and show that the convergence of $\lambda\mapsto Z_n(\lambda)/\lambda^n$ to $\lambda\mapsto W(\lambda)$ holds in the space of càdlàg functions on [0,\infty), equipped with Skorokhod's topology on compact sets.

Math and medical clinics: can mathematical modeling help in prognostics?Online: attend (Code 101816)

Cancer is a disease group annually causing nearly 10 million deaths worldwide. Treatment possibilities have increased over the past few years for numerous cancer subtypes. Still, quantitative prognostic frameworks are sparse and yet unestablished in clinical routine. This talk discusses a fully deterministic mathematical model capable to describe clinical cases quantitatively. Special attention will be given to its astonishing clinical benefit in prognostics already at very early disease stages.

Link and Passcode: https://tum-conf.zoom.us/j/96536097137 Code 101816

Effective constructions in algebraic topology with applications to machine learning and condensed matter physicsOnline: attend

It has long been envisioned that the strength of the barcode invariant could be increased using cohomology operations. Leveraging recent advances in the computation of Steenrod squares, we introduce a new family of computable invariants on mod 2 persistent cohomology termed $Sq^k$-barcodes. We present a complete algorithmic pipeline for their computation and illustrate their real-world applicability using the space of conformations of the cyclo-octane molecule. Time permitting, we will discuss further cochain level structures relevant to the classification of topological phases of matter.

Link: https://tum-conf.zoom.us/j/99397204146?pwd=VkVXMFVQUXRtVEt2RVJjMVNGR0RBQT09

Seminar page: https://wiki.tum.de/display/topology/Applied+Topology+Seminar

Some thoughts on modeling temporal communication networks and dynamical processesvia ZOOM (Boltzmannstr. 3, 85748 Garching)

In this talk I introduce a framework for modeling temporal communication networks and dynamical processes unfolding on such networks. The framework originates from the (new) observation that there is a meaningful division of temporal communication networks into six dynamic classes, where the class of a network is determined by its generating process. In particular, each class is characterized by a fundamental structure: a temporal-topological network motif, which corresponds to the network structure of communication events in that class of network. The fundamental structures constrain network configurations: only certain configurations are possible within a dynamic class. In this way the framework presented here highlights strong constraints on network structures, which simplify analyses and shape network flows. Therefore the fundamental structures hold the potential to impact how we model temporal networks.