Approximating $f$-Divergences with Rank Statistics

• URL: http://arxiv.org/abs/2601.22784v1
• Date: Fri, 30 Jan 2026 10:05:33 GMT
• Title: Approximating $f$-Divergences with Rank Statistics
• Authors: Viktor Stein, José Manuel de Frutos,
• Abstract summary: We introduce a rank-statistic approximation of $f$-divergences that avoids explicit density-ratio estimation by working directly with the distribution of ranks.<n>We prove that the resulting estimator of the divergence is monotone in $K$, is always a lower bound of the true $f$-divergence.<n>We empirically validate the approach by benchmarking against neural baselines and illustrating its use as a learning objective in generative modelling experiments.
• Score: 0.3222802562733787
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a rank-statistic approximation of $f$-divergences that avoids explicit density-ratio estimation by working directly with the distribution of ranks. For a resolution parameter $K$, we map the mismatch between two univariate distributions $μ$ and $ν$ to a rank histogram on $\{ 0, \ldots, K\}$ and measure its deviation from uniformity via a discrete $f$-divergence, yielding a rank-statistic divergence estimator. We prove that the resulting estimator of the divergence is monotone in $K$, is always a lower bound of the true $f$-divergence, and we establish quantitative convergence rates for $K\to\infty$ under mild regularity of the quantile-domain density ratio. To handle high-dimensional data, we define the sliced rank-statistic $f$-divergence by averaging the univariate construction over random projections, and we provide convergence results for the sliced limit as well. We also derive finite-sample deviation bounds along with asymptotic normality results for the estimator. Finally, we empirically validate the approach by benchmarking against neural baselines and illustrating its use as a learning objective in generative modelling experiments.

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