We discuss approximation of functions using deep neural nets. Given a function f on a d-dimensional manifold Γ⊂Rm, 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 Γ 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).
Bibliographical noteFunding Information:
The authors thank Stefan Steinerberger, Roy Lederman for their help, and to Andrew Barron, Ed Bosch, Mark Tygert and Yann LeCun for their comments. Alexander Cloninger is supported by NSF Award No. DMS-1402254 .
© 2016 Elsevier Inc.
- Function approximation
- Neural nets