Finding Local Dependent Regions in PDFs using RKHS Uncertainty Moments and Optimal Transport

Reliable measurement of dependence between random variables is essential in many applications of statistics and machine learning. Current approaches for dependence estimation employ the full probability density function (PDF) and are unable to quantify local dependence, which is required to improve precision, robustness and/or interpretability. We propose a two-step approach for local dependence quantification between random variables: 1) First decompose the PDF of the variables involved in orthogonal regional moments; 2) Compute an optimal transport map to measure the similarity, in the space of the data, between the corresponding sets of local low density moments, which correspond to uncertainty. Statistical dependence is then determined by the degree of one-to-one correspondence between the respective uncertainty moments decomposition. The proposed approach is robust towards outliers and monotone transformations of data, while the multiple moments of uncertainty provide high resolution and interpretability of the type of dependence being quantified. We support these claims through preliminary results using simulated data.