Scalable Mutual Information Estimation using Dependence Graphs
Abstract
The Mutual Information (MI) is an often used measure of dependency between two random variables utilized in information theory, statistics and machine learning. Recently several MI estimators have been proposed that can achieve parametric MSE convergence rate. However, most of the previously proposed estimators have the high computational complexity of at least $O(N^2)$. We propose a unified method for empirical nonparametric estimation of general MI function between random vectors in $\mathbb{R}^d$ based on $N$ i.i.d. samples. The reduced complexity MI estimator, called the ensemble dependency graph estimator (EDGE), combines randomized locality sensitive hashing (LSH), dependency graphs, and ensemble biasreduction methods. We prove that EDGE achieves optimal computational complexity $O(N)$, and can achieve the optimal parametric MSE rate of $O(1/N)$ if the density is $d$ times differentiable. To the best of our knowledge EDGE is the first nonparametric MI estimator that can achieve parametric MSE rates with linear time complexity. We illustrate the utility of EDGE for the analysis of the information plane (IP) in deep learning. Using EDGE we shed light on a controversy on whether or not the compression property of information bottleneck (IB) in fact holds for ReLu and other rectification functions in deep neural networks (DNN).
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1801.09125
 Bibcode:
 2018arXiv180109125N
 Keywords:

 Computer Science  Information Theory;
 Statistics  Machine Learning
 EPrint:
 19 Pages