Adaptive Independence Tests with Geo-Topological Transformation

Testing two potentially multivariate variables for statistical dependence on the basis finite samples is a fundamental statistical challenge. Here we explore a family of tests that adapt to the complexity of the relationship between the variables, promising robust power across scenarios. Building on the distance correlation, we introduce a family of adaptive independence criteria based on nonlinear monotonic transformations of distances. We show that these criteria, like the distance correlation and RKHS-based criteria, provide dependence indicators. We propose a class of adaptive (multi-threshold) test statistics, which form the basis for permutation tests. These tests empirically outperform some of the established tests in average and worst-case statistical sensitivity across a range of univariate and multivariate relationships and might deserve further exploration.