Provable approximation properties for deep neural networks

Uri Shaham; Alexander Cloninger; Ronald R. Coifman

doi:10.1016/j.acha.2016.04.003

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arXiv · 1509.07385

Provable approximation properties for deep neural networks

Uri Shaham·Alexander Cloninger·Ronald R. Coifman

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Abstract

We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $\Gamma \subset \mathbb{R}^m$, we construct a sparsely-connected depth-4 neural network and bound its error in approximating $f$. The size of the network depends on dimension and curvature of the manifold $\Gamma$, the complexity of $f$, in terms of its wavelet description, and only weakly on the ambient dimension $m$. Essentially, our network computes wavelet functions, which are computed from Rectified Linear Units (ReLU)

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Uri Shaham, Alexander Cloninger, Ronald R. Coifman. 2015-09-24. Provable approximation properties for deep neural networks. https://doi.org/10.1016/j.acha.2016.04.003

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