Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in $L^2(\mathbb{R}^d,\gamma_d)$

• URL: http://arxiv.org/abs/2111.07080v1
• Date: Sat, 13 Nov 2021 09:54:32 GMT
• Title: Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in $L^2(\mathbb{R}^d,\gamma_d)$
• Authors: Christoph Schwab and Jakob Zech
• Abstract summary: We prove expression rates for analytic functions $f:mathbbRdtomathbbR$ in the norm of $L2(mathbbRd,gamma_d)$. We consider in particular ReLU and ReLU$k$ activations for integer $kgeq 2$. As an application, we prove expression rate bounds of deep ReLU-NNs for response surfaces of elliptic PDEs with log-Gaussian random field inputs.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 ReLU and 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 $\mathbb{R}^{\mathbb{N}}$, 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 $\mathbb{C}^d$. As an application, we prove expression rate bounds of deep ReLU-NNs for response surfaces of elliptic PDEs with log-Gaussian random field inputs.

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