A multivariate Riesz basis of ReLU neural networks

Cornelia Schneider; Jan Vybíral

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arXiv · 2303.00076

A multivariate Riesz basis of ReLU neural networks

Cornelia Schneider·Jan Vybíral

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Abstract

We consider the trigonometric-like system of piecewise linear functions introduced recently by Daubechies, DeVore, Foucart, Hanin, and Petrova. We provide an alternative proof that this system forms a Riesz basis of $L_2([0,1])$ based on the Gershgorin theorem. We also generalize this system to higher dimensions $d>1$ by a construction, which avoids using (tensor) products. As a consequence, the functions from the new Riesz basis of $L_2([0,1]^d)$ can be easily represented by neural networks. Moreover, the Riesz constants of this system are independent of $d$, making it an attractive building block regarding future multivariate analysis of neural networks.

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Cornelia Schneider, Jan Vybíral. 2023-02-28. A multivariate Riesz basis of ReLU neural networks. https://arxiv.org/abs/2303.00076

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