Quasi-Monte Carlo for Multivariate Distributions via Generative Neural Networks
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Description
A novel approach based on generative neural networks is introduced for constructing quasi-random number generators for multivariate models with any underlying copula in order to estimate expectations with variance reduction. So far, quasi-random number generators for multivariate distributions required a careful design, exploiting specific properties (such as conditional distributions) of the implied copula or the underlying quasi-Monte Carlo point set, and were only tractable for a small number of models. Utilizing specific generative neural networks allows one to construct quasi-random number generators for a much larger variety of multivariate distributions without such restrictions. Once trained with a pseudo-random sample, these neural networks only require a multivariate standard uniform randomized quasi-Monte Carlo point set as input and are thus fast in estimating expectations under dependence with variance reduction. Reproducible numerical examples are considered to demonstrate the approach. Emphasis is put on ideas rather than mathematical proofs.
About Marius Hofert
Marius Hofert is an Associate Professor of Statistics in the Department of Statistics and Actuarial Science at University of Waterloo, Canada. He obtained his PhD in Mathematics from University of Ulm in 2010. He then held a postdoctoral research position at RiskLab, ETH Zürich. Before joining University of Waterloo, he had a guest professorship in the Department of Mathematics at Technische Universität München and a visiting assistant professorship in the Department of Applied Mathematics at University of Washington, Seattle. Marius' research interests are Computational Statistics and Data Science (data visualization, parallel computing, software development in R), Dependence Modeling with Copulas (high dimensional problems, hierarchical models, random number generation, computational aspects, graphical approaches) and Quantitative Risk Management (risk aggregation, risk measures, computational challenges).