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.

翻译：我们发展了互信息（MI）——信息论中一种成熟度量——在深度学习模型内部机制解释中的应用。为从有限样本中准确估计互信息，我们提出了GMM-MI（发音为"Jimmie"），一种基于高斯混合模型的算法，可同时适用于离散和连续场景。GMM-MI具有计算高效、对超参数选择鲁棒的特点，并能提供因有限样本量导致的互信息估计不确定性。我们在已知真实互信息的模拟数据上全面验证了GMM-MI，将其性能与现有互信息估计器进行比较。随后，我们展示了所提出的互信息估计器在表征学习中的应用，使用合成数据和描述高度非线性过程的物理数据集进行实验。我们训练深度学习模型将高维数据编码为有意义的压缩（潜）表征，并利用GMM-MI量化潜变量间的解缠程度及其与相关物理量的关联，从而解锁潜表征的可解释性。我们将GMM-MI公开于众。