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

arXiv: https://arxiv.org/abs/2004.04767 · HTML: https://mlsanity.com/paper/2004.04767/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2020.

## Built on by (most cited first, 4 of 4)

- [Wide and Deep Neural Networks Achieve Optimality for Classification](https://mlsanity.com/paper/2204.14126/) (2022)
- [Deep Networks Provably Classify Data on Curves](https://mlsanity.com/paper/2107.14324/) (2021)
- [Random features and polynomial rules](https://mlsanity.com/paper/2402.10164/) (2024)
- [Probability-Generating Function Kernels for Spherical Data](https://mlsanity.com/paper/2112.00365/) (2021)

## Abstract (arXiv)

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.