# Approximating f-Divergences with Rank Statistics

arXiv: https://arxiv.org/abs/2601.22784 · HTML: https://mlsanity.com/paper/2601.22784/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 60 references that are in the map.
- First posted on arXiv 30 Jan 2026.

## Builds on (most cited first, 26 of 26)

- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Assessing Generative Models via Precision and Recall](https://mlsanity.com/paper/1806.00035/) (2018)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Mitigating Bias in Calibration Error Estimation](https://mlsanity.com/paper/2012.08668/) (2020)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018)
- [Sliced Wasserstein Generative Models](https://mlsanity.com/paper/1904.05408/) (2019)
- [Generative Multi-Adversarial Networks](https://mlsanity.com/paper/1611.01673/) (2016)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [Practical and Consistent Estimation of f-Divergences](https://mlsanity.com/paper/1905.11112/) (2019)
- [Non-Asymptotic Performance Guarantees for Neural Estimation of mathsff-Divergences](https://mlsanity.com/paper/2103.06923/) (2021)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [MCL-GAN: Generative Adversarial Networks with Multiple Specialized Discriminators](https://mlsanity.com/paper/2107.07260/) (2021)
- [Revisiting Precision and Recall Definition for Generative Model Evaluation](https://mlsanity.com/paper/1905.05441/) (2019)
- [Precision-Recall Divergence Optimization for Generative Modeling with GANs and Normalizing Flows](https://mlsanity.com/paper/2305.18910/) (2023)
- [Multivariate f-Divergence Estimation With Confidence](https://mlsanity.com/paper/1411.2045/) (2014)
- [On the Theoretical Equivalence of Several Trade-Off Curves Assessing Statistical Proximity](https://mlsanity.com/paper/2006.11809/) (2020)
- [Training Implicit Generative Models via an Invariant Statistical Loss](https://mlsanity.com/paper/2402.16435/) (2024)
- [Robust training of implicit generative models for multivariate and heavy-tailed distributions with an invariant statistical loss](https://mlsanity.com/paper/2410.22381/) (2024)
- [Explicit Density Approximation for Neural Implicit Samplers Using a Bernstein-Based Convex Divergence](https://mlsanity.com/paper/2506.04700/) (2025)

## Abstract (arXiv)

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 nu 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→∞ 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 modeling experiments.