# On the identifiability of mixtures of ranking models

arXiv: https://arxiv.org/abs/2201.13132 · HTML: https://mlsanity.com/paper/2201.13132/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 57 references that are in the map.
- First posted on arXiv 31 Jan 2022.

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

- [Learning Mixtures of Linear Regressions with Nearly Optimal Complexity](https://mlsanity.com/paper/1802.07895/) (2018)
- [Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons](https://mlsanity.com/paper/1504.07218/) (2015)
- [Learning Mixtures of Plackett-Luce Models](https://mlsanity.com/paper/1603.07323/) (2016)
- [Efficiently Learning Mixtures of Mallows Models](https://mlsanity.com/paper/1808.05731/) (2018)
- [Data-driven Rank Breaking for Efficient Rank Aggregation](https://mlsanity.com/paper/1601.05495/) (2016)
- [Learning Mixed Multinomial Logit Model from Ordinal Data](https://mlsanity.com/paper/1411.0073/) (2014)
- [Learning Mixtures of Permutations: Groups of Pairwise Comparisons and Combinatorial Method of Moments](https://mlsanity.com/paper/2009.06784/) (2020)
- [Clustering and Inference From Pairwise Comparisons](https://mlsanity.com/paper/1502.04631/) (2015)
- [Learning an arbitrary mixture of two multinomial logits](https://mlsanity.com/paper/2007.00204/) (2020)
- [Composite Marginal Likelihood Methods for Random Utility Models](https://mlsanity.com/paper/1806.01426/) (2018)

## Built on by (most cited first, 7 of 7)

- [Distortion of AI Alignment: Does Preference Optimization Optimize for Preferences?](https://mlsanity.com/paper/2505.23749/) (2025)
- [Direct Alignment with Heterogeneous Preferences](https://mlsanity.com/paper/2502.16320/) (2025)
- [Clone-Robust AI Alignment](https://mlsanity.com/paper/2501.09254/) (2025)
- [Sharp analysis of EM for learning mixtures of pairwise differences](https://mlsanity.com/paper/2302.10066/) (2023)
- [A Revealed Preference Framework for AI Alignment](https://mlsanity.com/paper/2603.27868/) (2026)
- [Distortion of AI Alignment Revisited: RLHF is a Decent Utilitarian Aligner](https://mlsanity.com/paper/2609.12651/) (2026)
- [CURATRON: Complete and Robust Preference Data for Rigorous Alignment of Large Language Models](https://mlsanity.com/paper/2403.02745/) (2024)

## Abstract (arXiv)

Mixtures of ranking models are standard tools for ranking problems. However, even the fundamental question of parameter identifiability is not fully understood: the identifiability of a mixture model with two Bradley-Terry-Luce (BTL) components has remained open. In this work, we show that popular mixtures of ranking models with two components (BTL, multinomial logistic models with slates of size 3, or Plackett-Luce) are generically identifiable, i.e., the ground-truth parameters can be identified except when they are from a pathological subset of measure zero. We provide a framework for verifying the number of solutions in a general family of polynomial systems using algebraic geometry, and apply it to these mixtures of ranking models to establish generic identifiability. The framework can be applied more broadly to other learning models and may be of independent interest.