  The paper presents a new copula based method for measuring dependence between
random variables. Our approach extends the Maximum Mean Discrepancy to the
copula of the joint distribution. We prove that this approach has several
advantageous properties. Similarly to Shannon mutual information, the proposed
dependence measure is invariant to any strictly increasing transformation of
the marginal variables. This is important in many applications, for example in
feature selection. The estimator is consistent, robust to outliers, and uses
rank statistics only. We derive upper bounds on the convergence rate and
propose independence tests too. We illustrate the theoretical contributions
through a series of experiments in feature selection and low-dimensional
embedding of distributions.
