  Measuring the dependence of data plays a central role in statistics and
machine learning. In this work, we summarize and generalize the main idea of
existing information-theoretic dependence measures into a higher-level
perspective by the Shearer's inequality. Based on our generalization, we then
propose two measures, namely the matrix-based normalized total correlation
($T_\alpha^*$) and the matrix-based normalized dual total correlation
($D_\alpha^*$), to quantify the dependence of multiple variables in arbitrary
dimensional space, without explicit estimation of the underlying data
distributions. We show that our measures are differentiable and statistically
more powerful than prevalent ones. We also show the impact of our measures in
four different machine learning problems, namely the gene regulatory network
inference, the robust machine learning under covariate shift and non-Gaussian
noises, the subspace outlier detection, and the understanding of the learning
dynamics of convolutional neural networks (CNNs), to demonstrate their
utilities, advantages, as well as implications to those problems. Code of our
dependence measure is available at: https://bit.ly/AAAI-dependence
