Filter off: No filter for categories

The chemical distance in random interlacements in the low-intensity regime2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

Random interlacements is a Poissonian soup of doubly-infinite random walk trajectories on Z^d, with a parameter u > 0 controlling the intensity of the Poisson point process. In a natural way, the model defines a percolation on the edges of Z^d with long-range correlations. We consider the time constant associated to the chemical distance in random interlacements at low intensity u > 0. It is conjectured that the time constant times u^{1/2} converges to the Euclidean norm, as u ↓ 0. In dimensions d ≥ 5, we prove a sharp upper bound and an almost sharp lower bound for the time constant as the intensity decays to zero. Joint work with Eviatar Procaccia and Ron Rosenthal.

Bayesian Integrals on Toric VarietiesBC1 2.02.11 (Parkring 11, 85748 Garching)

Toric varieties have a strong combinatorial flavor: those algebraic varieties are described in terms of a fan. Based on joint work with M. Borinsky, B. Sturmfels, and S. Telen (https://arxiv.org/abs/2204.06414), I explain how to understand toric varieties as probability spaces. Bayesian integrals for discrete statistical models that are parameterized by a toric variety can be computed by a tropical sampling method. Our methods are motivated by the study of Feynman integrals and positive geometries in particle physics.

The Shift-Dimension: An Algebraic Invariant of Multiparameter Persistence ModulesMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

In the one-parameter case, persistence modules naturally are graded modules over the univariate polynomial ring and hence perfectly understood from an algebraic point of view. By a classical structure theorem, one associates the so-called “barcode”, from which one reads topological features of the data. Generalizing persistent homology to a multivariate setting allows for the extraction of finer information from data, but its algebraic properties are more subtle. In this talk, I discuss the shift-dimension, a stable invariant of multipersistence modules which is obtained as the hierarchical stabilization of a classical invariant. This talk is based on recent work ( https://arxiv.org/abs/2112.06509 ) with Wojciech Chachólski and René Corbet.

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

Stochastic resonance in stochastic PDEsOnline: attendvia ZOOM (Boltzmannstr. 3, 85748 Garching)

We consider stochastic partial differential equations (SPDEs) on the one-dimensional torus, driven by space-time white noise, and with a time-periodic drift term, which vanishes on two stable and one unstable equilibrium branches. Each of the stable branches approaches the unstable one once per period. We prove that there exists a critical noise intensity, depending on the forcing period and on the minimal distance between equilibrium branches, such that the probability that solutions of the SPDE make transitions between stable equilibria is exponentially small for subcritical noise intensity, while they happen with probability exponentially close to $1$ for supercritical noise intensity. Concentration estimates of solutions are given in the $H^s$ Sobolev norm for any $s<\frac12$. The results generalise to an infinite-dimensional setting those obtained for $1$-dimensional SDEs.

Optimal Discriminant Analysis in High-Dimensional Latent Factor ModelsOnline: attendBC1 2.01.10 (Parkring 11, 85748 Garching)

In high-dimensional classification problems, a commonly used approach is to first project the high-dimensional features into a lower dimensional space, and base the classification on the resulting lower dimensional projections. In this talk, we formulate a latent-variable model with a hidden low-dimensional structure to justify this two-step procedure and to guide which projection to choose. We propose a computationally efficient classifier that takes certain principal components (PCs) of the observed features as projections, with the number of retained PCs selected in a data-driven way. A general theory is established for analyzing such two-step classifiers based on any low-dimensional projections. We derive explicit rates of convergence of the excess risk of the proposed PC-based classifier. The obtained rates are further shown to be optimal up to logarithmic factors in the minimax sense. Our theory allows, but does not require, the lower-dimension to grow with the sample size and the feature dimension exceeds the sample size. Simulations support our theoretical findings. This is joint work with Xin Bing (Department of Statistical Sciences, University of Toronto).

Surprises in topic model estimation and new Wasserstein document-distance calculationsOnline: attend (Meeting-ID: 913-2473-4411; Password: StatsCol22)Seminarraum (Ludwigstrasse 33, 80333 Mathematisches Institut, LMU)

Topic models have been and continue to be an important modeling tool for an ensemble of independent multinomial samples with shared commonality. Although applications of topic models span many disciplines, the jargon used to deﬁne them stems from text analysis. In keeping with the standard terminology, one has access to a corpus of n independent documents, each utilizing words from a given dictionary of size p. One draws N words from each document and records their respective count, thereby representing the corpus as a collection of n samples from independent, p-dimensional, multinomial distributions, each having a diﬀerent, document speciﬁc, true word probability vector Π. The topic model assumption is that each Π is a mixture of K discrete distributions, that are common to the corpus, with document speciﬁc mixture weights. The corpus is assumed to cover K topics, that are not directly observable, and each of the K mixture components correspond to conditional probabilities of words, given a topic. The vector of the K mixture weights, per document, is viewed as a document speciﬁc topic distribution T, and is thus expected to be sparse, as most documents will only cover a few of the K topics of the corpus.

Despite the large body of work on learning topic models, the estimation of sparse topic distributions, of unknown sparsity, especially when the mixture components are not known, and are estimated from the same corpus, is not well understood and will be the focus of this talk. We provide estimators of T, with sharp theoretical guarantees, valid in many practically relevant situations, including the scenario p >> N (short documents, sparse data) and unknown K. Moreover, the results are valid when dimensions p and K are allowed to grow with the sample sizes N and n.

When the mixture components are known, we propose MLE estimation of the sparse vector T, the analysis of which has been open until now. The surprising result, and a remarkable property of the MLE in these models, is that, under appropriate conditions, and without further regularization, it can be exactly sparse, and contain the true zero pattern of the target. When the mixture components are not known, we exhibit computationally fast and rate optimal estimators for them, and propose a quasi-MLE estimator of T, shown to retain the properties of the MLE. The practical implication of our sharp, ﬁnite-sample, rate analyses of the MLE and quasi-MLE reveal that having short documents can be compensated for, in terms of estimation precision, by having a large corpus.

Our main application is to the estimation of Wasserstein distances between document generating distributions. We propose, estimate and analyze Wasserstein distances between alternative probabilistic document representations, at the word and topic level, respectively. The eﬀectiveness of the proposed Wasserstein distance estimates, and contrast with the more commonly used Word Mover Distance between empirical frequency estimates, is illustrated by an analysis of an IMDb movie reviews data set.

Brief Bio: Florentina Bunea obtained her Ph.D. in Statistics at the University of Washington, Seattle. She is now a Professor of Statistics in the Department of Statistics and Data Science, and she is affiliated with the Center for Applied Mathematics and the Department of Computer Science, at Cornell University. She is a fellow of the Institute of Mathematical Statistics, and she is or has been part of numerous editorial boards such as JRRS-B, JASA, Bernoulli, the Annals of Statistics. Her work has been continuously funded by the US National Science Foundation. Her most recent research interests include latent space models, topic models, and optimal transport in high dimensions.

Unifying information propagation models on networks and influence maximisationOnline: attendvia ZOOM (Boltzmannstr. 3, 85748 Garching)

Information propagation on networks is a central theme in social, behavioural, and economic sciences, with important theoretical and practical implications, such as the influence maximisation problem for viral marketing. Here, we consider a model that unifies the classical independent cascade models and the linear threshold models, and generalise them by considering continuous variables and allowing feedback in the dynamics. We then formulate its influence maximisation as a mixed integer nonlinear programming problem and adopt derivative-free methods. Furthermore, we show that the problem can be exactly solved in the special case of linear dynamics, where the selection criteria is closely related to the Katz centrality, and propose a customised direct search method with local convergence. We then demonstrate the close-to-optimal performance of the customised direct search numerically on both synthetic and real networks.

Overview of methods and results in Validated NumericsMI 02.10.011 (Boltzmannstr. 3, 85748 Garching)

This talk is aimed as an introduction into the topic of validated numerics, structured around an overview and justification of this exciting new branch in numerical analysis. At its core, validated numerics encompasses the mathematical proofs developed with and thanks to a computer. The name of the game is then to use the power of a computer to let it handle most computational effort while the mathematician structures the proof. We will see how interval arithmetics is at the cornerstone of this search. Then, we will concentrate on dynamics, showing how different methods, either based on functional analysis (radii polynomial approach, homotopy method), or more geometrical interpretations (CAPD), have tackled many problems in various and ingenious ways.

We will finish with a picture gallery of results, showing the flexibility and reach of validated numerics

Sufficient optimality conditions in optimization and optimal controlGebäude 33, Raum 1401 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

Sufficient optimality conditions are very important for several reasons: First, they are the main ingredient for stability of perturbed problems. Under certain assumptions one can show Lipschitz stability of perturbed solutions. Second, such conditions are essentially needed to show fast convergence of SQP-methods. Third, sufficient optimality conditions are a main tool to derive optimal a priori error estimates for FE-discretizations for nonconvex optimal control problems.

We will start with an overview on second-order optimality conditions for finite dimensional optimization problems. This will be the starting point to derive no-gap sufficient optimality conditions for optimal control problems.

On signatures of families of manifolds Online: attend (Meeting-ID: 913-2473-4411; Password: StatsCol22)A 027 (Theresienstr. 39, 80333 München)

Digits and dynamics - A brief tour of Benford's LawMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Benford's Law, a notorious gem of mathematics folklore, asserts that significant digits of numerical data are not usually equidistributed, as might be expected, but rather follow one particular logarithmic distribution. Since first recorded by Newcomb in 1881, this apparently counter-intuitive phenomenon has attracted much interest from scientists and mathematicians alike. This talk will introduce and discuss several intriguing aspects of the phenomenon, relating them to problems in stochastics, number theory and, above all, dynamics.

A Variational Ansatz for the Quantum Sherrington-Kirkpatrick ModelMI 02.10.011 (Boltzmannstr. 3, 85748 Garching)

I will present some results that can be obtained thanks to a new variational study of the Quantum Sherrington-Kirkpatrick (QSK) model. First I will introduce a variational ansatz (which we refer to as generalised spin coherent states) and discuss how it can be optimised to describe the QSK ground state, leading to a better description compared to other possible ansätze. Then I will show how this description correctly captures quantitative properties of the model such as the ground state energy and the spin glass phase transition. This explicit description of the ground state wavefunction allows us to paint a simple picture of its structure (which we can understand as an ensemble of states defined by normally distributed two-body gates) and compute some of its fundamental properties such as its entanglement (which we find follows a volume law behaviour).

*-Edge-reinforced random walk and *-vertex-reinforced jump processes: limit measure and random Schrödinger representation2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

TBA

Estimation in Cox proportional hazards model with measurement errorsOnline: attendBC1 2.01.10 (Parkring 11, 85748 Garching)

The Cox proportional hazards model is a semiparametric regression model that can be used in medical research, engineering or insurance for investigating the association between the survival time (the so-called lifetime) of an object and predictor variables. We investigate the Cox proportional hazards model for right-censored data, where the baseline hazard rate belongs to an unbounded set of nonnegative Lipschitz functions, with fixed constant, and the vector of regression parameters belongs to a compact parameter set, and in addition, the time-independent covariates are subject to measurement errors. We construct a simultaneous estimator of the baseline hazard rate and regression parameter, present asymptotic results and discuss goodness-of-fit tests.

Bi-Objective Reliability Based Design Optimization: Applications in Portfolio InvestmentB349 (Theresienstraße 39, 80333 Mathematisches Institut, LMU)

The seminal work of Markowitz (1952) paved the way by opening up the exciting field of portfolio optimization to future researchers, whereby different variants of financial optimization models have been solved, with interesting results. Portfolio optimization problems are easy to address considering linear objective functions, subject to different risks, returns and investment constraints, under the assumption of normal distribution of asset returns. Higher complexity arises if (i) non-linear assets, (ii) non-normal distribution of asset returns along with (iii) uncertainty in parameter estimates under a (iv) multi-objective framework is considered. In this paper we solve two interesting variants of multi-objective financial optimization problems by considering non-normal asset return distributions along with uncertainty in parameter estimates. Pre-processing of the input data for optimization is accomplished using relevant ARCH and GARCH techniques applied on returns which are EVD. The efficacies of our proposed multiobjective reliability based portfolio optimization (MORBPO) problems, solved utilizing Genetic Algorithm (GA), and self developed Python codes (for both optimization and reliability calculations) are demonstrated using data from the Indian financial markets. We present the detailed results pertaining to the optimal values of weights of investments, investment returns, variance, CVaR, EVaR, reliability index levels (β), efficient Pareto optimal frontiers, and analyze the practical implications of these, before concluding the paper with some motivating future research ideas

Authors: Raghu Nandan Sengupta, Aditya GUPTA, Subhankar MUKHERJEE and Gregor WEISS

Modeling Implied Volatility Surfaces of Crypto OptionsB349 (Theresienstraße 39, 80333 Mathematisches Institut, LMU)

Modelling of implied volatility surfaces of crypto assets, such as Bitcoin and Ether, is challenging when applying conventional stochastic volatility (SV) models. On the theoretical side, the dominance of inverse pay-off functions requires stronger conditions for the existence of equivalent martingale measures. On the practical side, the positive correlation between the price-returns and volatility and price may either invalidate some SV models or pose tight restrictions on model parameters. First, we introduce a market model for the valuation of vanilla and inverse options on crypto assets under the equivalent martingale money market account measure and the inverse spot measure. We derive conditions for the existence of such valuation measures under conventional SV models and we show limitations of popular SV models when the correlation between returns and their volatility dynamics is positive. Second, we introduce the lognormal SV model with the quadratic drift for arbitrage-free dynamics of implied volatilities. Despite the non-linear drift, we show that the volatility process has a strong solution. We derive the domain model parameters for which the equivalent spot and inverse measures exists. We then develop an analytic approach for valuation of options on price and quadratic variance based on a new method of affine expansion for the moment generation function of price processes with non-affine and non-linear dynamics. We proof the stability of the expansion method and demonstrate its accuracy in option pricing applications. We finally illustrate the application of the lognormal SV model for calibrating and modeling of Bitcoin and Ether volatility surfaces using Deribit options exchange data. This talk is based on joint work with Parviz Rakhmonov.

Web3 and the Future of BankingB349 (Theresienstraße 39, 80333 Mathematisches Institut, LMU)

Crypto and Blockchain have been around for quite a few years. The first wave of adoption was led by trading companies like centralized exchanges as crypto currencies became a means of speculation. Still. Blockchain was invented for transacting money or value. Nowadays there is a second wave happening with Decentralized Finance. Blockchains are now used to deliver financial services end-to-end in a new decentralized way. Web3 as a broader topic offers a big opportunity to make digital ecosystems converge, moving finance, identity, social networks and creator economy out of their silos, to become one big global digital decentralized economy. The talk gives insights to the current state of development and how Decentralized Finance will build the foundation for the best financial service infrastructure in human history.

Data-driven analysis of complex dynamical systemsMI 02.10.011 (Boltzmannstr. 3, 85748 Garching)

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We will present kernel-based methods for the approximation of transfer operators and differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. We will illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics, fluid dynamics, and quantum mechanics.