  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, offer
useful insights to the data and may deserve further exploration.
