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Generative neural networks for characteristic functions

Florian Brück

TL;DR

A simulation algorithm to simulate from a (multivariate) characteristic function, which is only accessible in a black-box format, is provided, based on a generative neural network, whose loss function exploits a specific representation of the Maximum-Mean-Discrepancy metric to directly incorporate the targeted characteristic function.

Abstract

We provide a simulation algorithm to simulate from a (multivariate) characteristic function, which is only accessible in a black-box format. The method is based on a generative neural network, whose loss function exploits a specific representation of the Maximum-Mean-Discrepancy metric to directly incorporate the targeted characteristic function. The algorithm is universal in the sense that it is independent of the dimension and that it does not require any assumptions on the given characteristic function. Furthermore, finite sample guarantees on the approximation quality in terms of the Maximum-Mean Discrepancy metric are derived. The method is illustrated in a simulation study.

Generative neural networks for characteristic functions

TL;DR

A simulation algorithm to simulate from a (multivariate) characteristic function, which is only accessible in a black-box format, is provided, based on a generative neural network, whose loss function exploits a specific representation of the Maximum-Mean-Discrepancy metric to directly incorporate the targeted characteristic function.

Abstract

We provide a simulation algorithm to simulate from a (multivariate) characteristic function, which is only accessible in a black-box format. The method is based on a generative neural network, whose loss function exploits a specific representation of the Maximum-Mean-Discrepancy metric to directly incorporate the targeted characteristic function. The algorithm is universal in the sense that it is independent of the dimension and that it does not require any assumptions on the given characteristic function. Furthermore, finite sample guarantees on the approximation quality in terms of the Maximum-Mean Discrepancy metric are derived. The method is illustrated in a simulation study.
Paper Structure (11 sections, 3 theorems, 22 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 11 sections, 3 theorems, 22 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Construction of the generator
  3. MMD metrics based on translation invariant kernels
  4. Construction of the loss function
  5. Theoretical guarantees
  6. Simulation results
  7. Conclusion and Discussion
  8. Conclusion
  9. Discussion
  10. Proof of Theorem \ref{['thmtheoreticalguarantees']}
  11. Additional plots

Key Result

Corollary 3.1

Let $\bm{X}$ denote an arbitrary random vector. Assume that $P_{\bm{Z}}$ is an absolutely continuous probability distribution w.r.t. the Lebesgue measure. Then there exists a sequence of fully connected neural networks $N_{{\bm{\theta}}_n}(\cdot)$ of depth $2$ with ReLu activation function such that

Figures (3)

  • Figure 1: Estimated marginal (left) and bivariate (right) densities of the Gaussian mixture distribution in dimensions $2,5,10$ with $2,5,10$ mixture components. The bivariate densities correspond to the last two components of the corresponding random vector. The solid red lines show the true densities and the dashed green and dash-dotted blue lines correspond to estimated densities from the generator and the exact simulation algorithm.
  • Figure 2: Exemplary visualization of the training loss during $6000$ epochs
  • Figure 3: Estimated marginal (left) and bivariate (right) densities of the Gaussian mixture distribution in dimensions $2,5,10$ with $2,5,10$ mixture components. The bivariate densities correspond to the last two components of the corresponding random vector. The solid red lines show the true densities and the dashed green and dash-dotted blue lines correspond to estimated densities from the generator and the exact simulation algorithm.

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Corollary 3.1
  • Theorem 3.2
  • Lemma 1
  • proof : Proof of Theorem \ref{['thmtheoreticalguarantees']}
  • proof : Proof of Lemma \ref{['lemlossfctisrandommmd']}