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Dettaglio Insegnamento

Contesto
Anno Accademico 2021/2022
Corso di Studi Dott. - MI (1385) Modelli e Metodi Matematici per l'Ingegneria / Mathematical Models and Methods in Engineering
Anno di Corso 1

Scheda Insegnamento
Codice Identificativo 057408
Denominazione Insegnamento WASSERSTEIN METRICS: FROM DIFFUSION EQUATIONS TO STATISTICAL ISSUES
Tipo Insegnamento MONODISCIPLINARE
Crediti Formativi Universitari (CFU) 5.0
Programma sintetico The Kantorovich-Wasserstein metric has been successfully used in several branches of mathe- matics, including probability, analysis and statistics. Moreover, it has recently found applications also in computer science, machine learning and image processing. The goal of this course is to provide an overview of the basic theory of the Kantorovich- Wasserstein metric and its wide context of application. The main topics will include a short introduction to probability metrics and to the theory of the Kantorovich-Wasserstein metrics, con- nections with the theory of linear (and possibly nonlinear) PDEs of diusion type seen as metric gradient ows, applications to quantitative central limit problems, computational issues and, time permitting, also to statistics (Wasserstein means and clustering). program: An introduction to probability metrics: semi-distances, simple and compound metrics, ideal metrics. [16] Kantorovich-Wasserstein functional: duality and basic properties. Connections with the Transportation Problem. [2, 20, 21] Weak convergence of measures and the topology induced by the Kantorovich-Wassersteinmetrics. The Wasserstein distance on the real line. [2, 16, 20, 21] Gradient flows generated by lower-semicontinuous convex functionals: from the classical theory in Rn to the EVI (Evolution Variational Inequality) formulation in a metric setting.[17, 18]. An overview of the differentiable structure of the Wasserstein space of probability measures in Rn. [2, 4] The heat equation in the Euclidean space as a gradient flow of the entropy functional in the Wasserstein space, and related stability properties. If time allows, possible extensions to more general (nonlinear) diffusion equations and to non-Euclidean frameworks. [2, 3, 4,7, 9, 13, 14]. Rate of convergence in central limit problems. [8, 15] Wasserstein Barycenters and some statistical applications. [15, 19] Computational issues in discrete setting: regularized transport and connection with flow on graphs. [1, 5, 6, 10, 11, 18, 19]
Settori Scientifico Disciplinari (SSD)
Codice SSD Descrizione SSD CFU
MAT/06 PROBABILITA' E STATISTICA MATEMATICA 2.5

Dettaglio
Scaglione Docente Programma dettagliato
Da (compreso) A (escluso)
A ZZZZ Bassetti Federico, Muratori Matteo
manifestidott v. 1.7.0 / 1.7.0
Area Servizi ICT
12/08/2022