Mehler’s Formula, Branching Process, and Compositional Kernels of Deep Neural Networks

42 Pages Posted: 19 Oct 2020

See all articles by Tengyuan Liang

Tengyuan Liang

University of Chicago - Booth School of Business

Hai Tran-Bach

University of Chicago

Date Written: October 16, 2020

Abstract

We utilize a connection between compositional kernels and branching processes via Mehler’s formula to study deep neural networks. This new probabilistic insight provides us a novel perspective on the mathematical role of activation functions in compositional neural networks. We study the unscaled and rescaled limits of the compositional kernels and explore the different phases of the limiting behavior, as the compositional depth increases. We investigate the memorization capacity of the compositional kernels and neural networks by characterizing the interplay among compositional depth, sample size, dimensionality, and non-linearity of the activation. Explicit formulas on the eigenvalues of the compositional kernel are provided, which quantify the complexity of the corresponding reproducing kernel Hilbert space. On the methodological front, we propose a new random features algorithm, which compresses the compositional layers by devising a new activation function.

Suggested Citation

Liang, Tengyuan and Tran-Bach, Hai, Mehler’s Formula, Branching Process, and Compositional Kernels of Deep Neural Networks (October 16, 2020). University of Chicago, Becker Friedman Institute for Economics Working Paper No. 2020-151, Available at SSRN: https://ssrn.com/abstract=3714014 or http://dx.doi.org/10.2139/ssrn.3714014

Tengyuan Liang (Contact Author)

University of Chicago - Booth School of Business ( email )

Hai Tran-Bach

University of Chicago ( email )

1101 East 58th Street
Chicago, IL 60637
United States

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
29
Abstract Views
370
PlumX Metrics