Decomposing neural networks as mappings of correlation functions

TL;DR

This paper analyzes how deep neural networks process information by mapping probability distributions through correlation functions, revealing that second-order correlations mainly drive internal layer processing and higher-order correlations are extracted at the input layer.

Contribution

It introduces a novel framework to decompose neural network functions into transformations of correlation functions, providing insights into the role of different order correlations in information processing.

Findings

Second-order correlations dominate internal layer processing.

Higher-order correlations are primarily extracted at the input layer.

The approach offers a quantitative perspective on neural network classification.

Abstract

Understanding the functional principles of information processing in deep neural networks continues to be a challenge, in particular for networks with trained and thus non-random weights. To address this issue, we study the mapping between probability distributions implemented by a deep feed-forward network. We characterize this mapping as an iterated transformation of distributions, where the non-linearity in each layer transfers information between different orders of correlation functions. This allows us to identify essential statistics in the data, as well as different information representations that can be used by neural networks. Applied to an XOR task and to MNIST, we show that correlations up to second order predominantly capture the information processing in the internal layers, while the input layer also extracts higher-order correlations from the data. This analysis provides a quantitative and explainable perspective on classification.