A robust estimator of mutual information for deep learning interpretability

TL;DR

This paper introduces GMM-MI, a robust and efficient mutual information estimator based on Gaussian mixture models, to interpret deep learning models by quantifying information in latent representations.

Contribution

We propose GMM-MI, a novel mutual information estimator suitable for both discrete and continuous data, with robustness, efficiency, and uncertainty quantification, validated on toy and real datasets.

Findings

GMM-MI accurately estimates MI compared to existing methods.

It effectively measures disentanglement and relevance in deep representations.

GMM-MI enhances interpretability of deep learning models.

Abstract

We develop the use of mutual information (MI), a well-established metric in information theory, to interpret the inner workings of deep learning models. To accurately estimate MI from a finite number of samples, we present GMM-MI (pronounced $``$Jimmie$"$), an algorithm based on Gaussian mixture models that can be applied to both discrete and continuous settings. GMM-MI is computationally efficient, robust to the choice of hyperparameters and provides the uncertainty on the MI estimate due to the finite sample size. We extensively validate GMM-MI on toy data for which the ground truth MI is known, comparing its performance against established mutual information estimators. We then demonstrate the use of our MI estimator in the context of representation learning, working with synthetic data and physical datasets describing highly non-linear processes. We train deep learning models to encode high-dimensional data within a meaningful compressed (latent) representation, and use GMM-MI to quantify both the level of disentanglement between the latent variables, and their association with relevant physical quantities, thus unlocking the interpretability of the latent representation. We make GMM-MI publicly available.