Bergsma--Dassios Sign Covariance Characterises Independence for Arbitrary Real-Valued Bivariate Laws

What the math gives to ML

The paper supplies a distribution-free, rank-based independence certificate that remains valid for arbitrary real-valued bivariate laws, including ties, mixed distributions, and singular support. Its key transferable asset is the quantitative relation \(\tau^{*}\ge 2\mathcal{B}\): driving the quartet statistic toward zero controls a Blum–Kiefer–Rosenblatt dependence functional rather than merely matching correlations or pairwise moments. A practical neural-network adaptation is a minibatch quartet regularizer computed with a differentiable soft-sign approximation, used to remove nonlinear dependence between a learned representation and a nuisance variable while preserving task information.