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Exponential ReLU DNN expression of holomorphic maps in high dimension

by J. A. A. Opschoor and Ch. Schwab and J. Zech

(Report number 2019-35)

Abstract
For a parameter dimension $d\in\mathbb{N}$, we consider the approximation of many-parametric maps $u: [-1,1]^d\to \mathbb{R}$ by deep ReLU neural networks. The input dimension $d$ may possibly be large, and we assume quantitative control of the domain of holomorphy of $u$: i.e., $u$ admits a holomorphic extension to a Bernstein polyellipse $\mathcal{E}_{\rho_1}\times ... \times \mathcal{E}_{\rho_d} \subset \mathbb{C}^d$ of semiaxis sums $\rho_i>1$ containing $[-1,1]^{d}$. We establish the exponential rate $O(\exp(-bN^{1/(d+1)}))$ of expressive power in terms of the total NN size $N$ and of the input dimension $d$ of the ReLU NN in $W^{1,\infty}([-1,1]^d)$. The constant $b>0$ depends on $(\rho_j)_{j=1}^d$ which characterizes the coordinate-wise sizes of the Bernstein-ellipses for $u$. We also prove exponential convergence in stronger norms for the approximation by DNNs with more regular, so-called ``rectified power unit'' (RePU) activations. Finally, we extend DNN expression rate bounds also to two classes of non-holomorphic functions, in particular to $d$-variate, Gevrey-regular functions, and, by composition, to certain multivariate probability distribution functions with Lipschitz marginals.

Keywords: Deep ReLU neural networks, approximation rates, exponential convergence

@Techreport{OSZ19_839,
  author = {J. A. A. Opschoor and Ch. Schwab and J. Zech},
  title = {Exponential ReLU DNN expression of holomorphic maps in high dimension},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-35},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-35.pdf },
  year = {2019}
}

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