The universal approximation power of finite-width deep ReLU networks

Authors

Dmytro Perekrestenko, Philipp Grohs, Dennis Elbrächter, and Helmut Bölcskei

Reference

Conference on Neural Information Processing Systems, June 2018, submitted.

Abstract

We show that finite-width deep ReLU neural networks yield rate-distortion optimal approximation (Bölcskei et al., 2018) of polynomials, windowed sinusoidal functions, one-dimensional oscillatory textures, and the Weierstrass function, a fractal function which is continuous but nowhere differentiable. Together with their recently established universal approximation property of affine function systems (Bölcskei et al., 2018), this shows that deep neural networks approximate vastly different signal structures generated by the affine group, the Weyl-Heisenberg group, or through warping, and even certain fractals, all with approximation error decaying exponentially in the number of neurons. We also prove that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.

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