# Statistical Depth Functions for Ranking Distributions: Definitions, Statistical Learning and Applications

arXiv: https://arxiv.org/abs/2201.08105 · HTML: https://mlsanity.com/paper/2201.08105/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 73 references that are in the map.
- First posted on arXiv 20 Jan 2022.

## Builds on (most cited first, 10 of 10)

- [Simple, Robust and Optimal Ranking from Pairwise Comparisons](https://mlsanity.com/paper/1512.08949/) (2015)
- [Stochastically Transitive Models for Pairwise Comparisons: Statistical and Computational Issues](https://mlsanity.com/paper/1510.05610/) (2015)
- [Efficiently Learning Mixtures of Mallows Models](https://mlsanity.com/paper/1808.05731/) (2018)
- [Optimal Learning of Mallows Block Model](https://mlsanity.com/paper/1906.01009/) (2019)
- [Exact computation of the halfspace depth](https://mlsanity.com/paper/1411.6927/) (2014)
- [Learning Mixtures of Plackett-Luce Models from Structured Partial Orders](https://mlsanity.com/paper/1910.11721/) (2019)
- [Depth for curve data and applications](https://mlsanity.com/paper/1901.00180/) (2019)
- [Concentric mixtures of Mallows models for top-k rankings: sampling and identifiability](https://mlsanity.com/paper/2010.14260/) (2020)
- [Depth and depth-based classification with R-package ddalpha](https://mlsanity.com/paper/1608.04109/) (2016)
- [Depth-based Runs Tests for Bivariate Central Symmetry](https://mlsanity.com/paper/1401.2377/) (2014)

## Built on by (most cited first, 5 of 5)

- [Comparing Machine Learning Algorithms by Union-Free Generic Depth](https://mlsanity.com/paper/2312.12839/) (2023)
- [Robust Consensus in Ranking Data Analysis: Definitions, Properties and Computational Issues](https://mlsanity.com/paper/2303.12878/) (2023)
- [Membership Inference Attacks via Adversarial Examples](https://mlsanity.com/paper/2207.13572/) (2022)
- [Beyond Kemeny Medians: Consensus Ranking Distributions Definition, Properties and Statistical Learning](https://mlsanity.com/paper/2602.10640/) (2026)
- [Decentralized Ranking Aggregation via Gossip: Convergence and Robustness](https://mlsanity.com/paper/2602.22847/) (2026)

## Abstract (arXiv)

The concept of median/consensus has been widely investigated in order to provide a statistical summary of ranking data, i.e. realizations of a random permutation Sigma of a finite set, 1,; ldots,; n with n≥ 1 say. As it sheds light onto only one aspect of Sigma's distribution P, it may neglect other informative features. It is the purpose of this paper to define analogs of quantiles, ranks and statistical procedures based on such quantities for the analysis of ranking data by means of a metric-based notion of depth function on the symmetric group. Overcoming the absence of vector space structure on mathfrakSn, the latter defines a center-outward ordering of the permutations in the support of P and extends the classic metric-based formulation of consensus ranking (medians corresponding then to the deepest permutations). The axiomatic properties that ranking depths should ideally possess are listed, while computational and generalization issues are studied at length. Beyond the theoretical analysis carried out, the relevance of the novel concepts and methods introduced for a wide variety of statistical tasks are also supported by numerous numerical experiments.