Approximating $f$-Divergences with Rank Statistics
We introduce a rank-statistic approximation of $f$-divergences that avoids explicit density-ratio estimation by working directly with rank distributions. For a resolution parameter $K$, the mismatch between two univariate distributions $\mu$ and $\nu$ is mapped to a rank histogram on $\{0,\ldots,K\}$, whose deviation from uniformity is measured through a discrete $f$-divergence. We show that this estimator is monotone in $K$, provides a lower bound on the true $f$-divergence, and converges quantitatively as $K\to\infty$ under mild regularity assumptions. We extend the construction to high dimensions through random projections, establish convergence, deviation, and asymptotic normality results, and validate the method empirically in divergence estimation and generative modelling.
Palabras clave: f-divergences rank statistics divergence estimation density-ratio-free methods