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The authors propose a nonlinear measure of dependence between two random variables. This turns out to be the canonical correlation between random, nonlinear projections of the variables after a copula transformation which renders the marginals of the r.vs invariant to linear transformations. The paper introduces a new method called RDC to measure the statistical dependence between random variables. It combines a copula transform to a variant of kernel CCA using random projections, resulting in a $O(n log n)$ complexity. Results on synthetic and real benchmark data show promising results for feature selection. The RDC is a nonlinear dependency estimator that satisfies Renyi's criteria and exploits the very recent FastFood speedup trick (ICML13) \cite{journals/corr/LeSS14}. This is a straightforward recipe: 1) copularize the data, effectively preserving the dependency structure while ignoring the marginals, 2) sample k nonlinear features of each datum (inspired from Bochner's theorem) and 3) solve the regular CCA eigenvalue problem on the resulting paired datasets. Ultimately, RDC feels like a copularised variation of kCCA (misleading as this may sound). Its efficiency is illustrated successfully on a set of classical nonlinear bivariate dependency scenarios and 12 real datasets via a forward feature selection procedure.
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