# Learning Mixtures of Permutations: Groups of Pairwise Comparisons and Combinatorial Method of Moments

arXiv: https://arxiv.org/abs/2009.06784 · HTML: https://mlsanity.com/paper/2009.06784/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2020.

## Built on by (most cited first, 6 of 6)

- [MallowsPO: Fine-Tune Your LLM with Preference Dispersions](https://mlsanity.com/paper/2405.14953/) (2024)
- [On the identifiability of mixtures of ranking models](https://mlsanity.com/paper/2201.13132/) (2022)
- [Robust Voting Rules from Algorithmic Robust Statistics](https://mlsanity.com/paper/2112.06380/) (2021)
- [Sharp analysis of EM for learning mixtures of pairwise differences](https://mlsanity.com/paper/2302.10066/) (2023)
- [Mallows Model with Learned Distance Metrics: Sampling and Maximum Likelihood Estimation](https://mlsanity.com/paper/2507.08108/) (2025)
- [Population-Level Generative Modeling for Ranking Data](https://mlsanity.com/paper/2608.08422/) (2026)

## Abstract (arXiv)

In applications such as rank aggregation, mixture models for permutations are frequently used when the population exhibits heterogeneity. In this work, we study the widely used Mallows mixture model. In the high-dimensional setting, we propose a polynomial-time algorithm that learns a Mallows mixture of permutations on n elements with the optimal sample complexity that is proportional to log n, improving upon previous results that scale polynomially with n. In the high-noise regime, we characterize the optimal dependency of the sample complexity on the noise parameter. Both objectives are accomplished by first studying demixing permutations under a noiseless query model using groups of pairwise comparisons, which can be viewed as moments of the mixing distribution, and then extending these results to the noisy Mallows model by simulating the noiseless oracle.