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Deep Learning in High Dimension: Neural Network Expression Rates for Analytic Functions in $L^2(\R^d,\gamma_d)$

by Ch. Schwab and J. Zech

(Report number 2021-40)

Abstract
For artificial deep neural networks, we prove expression rates for analytic functions \(f:{\mathbb R}^d\to {\mathbb R}\) in the norm of \(L^2({\mathbb R}^d,\gamma_d)\) where \(d\in {\mathbb N}\cup\{ \infty \}\). Here \(\gamma_d\) denotes the Gaussian product probability measure on \({\mathbb R}^d\). We consider in particular \({\mathrm{ReLU}}\) and \({\mathrm{ReLU}}^k\) activations for integer \(k\geq 2\). For \(d\in\mathbb{N}\), we show exponential convergence rates in \(L^2(\mathbb{R}^d,\gamma_d)\). In case \(d=\infty\), under suitable smoothness and sparsity assumptions on \(f:{\mathbb R}^{\mathbb N}\to {\mathbb R}\), with \(\gamma_\infty\) denoting an infinite (Gaussian) product measure on \(({\mathbb R}^{\mathbb N}, {\mathcal B}({\mathbb R}^{\mathbb N}))\), we prove dimension-independent expression rate bounds in the norm of \(L^2({\mathbb R}^{\mathbb N},\gamma_\infty)\). The rates only depend on quantified holomorphy of (an analytic continuation of) the map \(f\) to a product of strips in \({\mathbb C}^d\) (in \({\mathbb C}^{\mathbb N}\) for \(d=\infty\), respectively). As an application, we prove expression rate bounds of deep \({\mathrm{ReLU}}\)-NNs for response surfaces of elliptic PDEs with log-Gaussian random field inputs.

Keywords:

@Techreport{SZ21_982,
  author = {Ch. Schwab and J. Zech},
  title = {Deep Learning in High Dimension: Neural Network Expression Rates for Analytic Functions in $L^2(\R^d,\gamma_d)$},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-40},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-40.pdf },
  year = {2021}
}

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