# A multivariate Riesz basis of ReLU neural networks

## TL;DR

This paper proves that a certain system of piecewise linear functions forms a Riesz basis for multivariate $L_2$ spaces, which can be efficiently represented by neural networks and is dimension-independent.

## Contribution

It provides an alternative proof of the Riesz basis property and generalizes the system to higher dimensions without tensor products, facilitating neural network representations.

## Key findings

- The system forms a Riesz basis of $L_2([0,1])$.
- The basis generalizes to higher dimensions $d>1$.
- Riesz constants are independent of the dimension $d$.

## 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.