# Spectral Ranking Inferences based on General Multiway Comparisons

arXiv: https://arxiv.org/abs/2308.02918 · HTML: https://mlsanity.com/paper/2308.02918/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 49 references that are in the map.
- First posted on arXiv 5 Aug 2023.

## Builds on (most cited first, 13 of 13)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence](https://mlsanity.com/paper/1505.01462/) (2015)
- [Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking](https://mlsanity.com/paper/1707.09971/) (2017)
- [Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons](https://mlsanity.com/paper/1504.07218/) (2015)
- [Uncertainty quantification in the Bradley-Terry-Luce model](https://mlsanity.com/paper/2110.03874/) (2021)
- [Dynamic Assortment Optimization with Changing Contextual Information](https://mlsanity.com/paper/1810.13069/) (2018)
- [Minimax-optimal Inference from Partial Rankings](https://mlsanity.com/paper/1406.5638/) (2014)
- [Ranking Inferences Based on the Top Choice of Multiway Comparisons](https://mlsanity.com/paper/2211.11957/) (2022)
- [Lagrangian Inference for Ranking Problems](https://mlsanity.com/paper/2110.00151/) (2021)
- [Uncertainty Quantification of MLE for Entity Ranking with Covariates](https://mlsanity.com/paper/2212.09961/) (2022)
- [Co-citation and Co-authorship Networks of Statisticians](https://mlsanity.com/paper/2204.11194/) (2022)
- [ℓ∞-Bounds of the MLE in the BTL Model under General Comparison Graphs](https://mlsanity.com/paper/2110.10825/) (2021)
- [A unified analysis of likelihood-based estimators in the Plackett--Luce model](https://mlsanity.com/paper/2306.02821/) (2023)

## Built on by (most cited first, 11 of 11)

- [Diffusion Model for Data-Driven Black-Box Optimization](https://mlsanity.com/paper/2403.13219/) (2024)
- [The ICML 2023 Ranking Experiment: Examining Author Self-Assessment in ML/AI Peer Review](https://mlsanity.com/paper/2408.13430/) (2024)
- [Covariate Assisted Entity Ranking with Sparse Intrinsic Scores](https://mlsanity.com/paper/2407.08814/) (2024)
- [Contextual Online Uncertainty-Aware Preference Learning for Human Feedback](https://mlsanity.com/paper/2504.19342/) (2025)
- [Statistical ranking with dynamic covariates](https://mlsanity.com/paper/2406.16507/) (2024)
- [Prompt-Dependent Ranking of Large Language Models with Uncertainty Quantification](https://mlsanity.com/paper/2603.03336/) (2026)
- [Top-K ranking with a monotone adversary](https://mlsanity.com/paper/2402.07445/) (2024)
- [Random pairing MLE for estimation of item parameters in Rasch model](https://mlsanity.com/paper/2406.13989/) (2024)
- [A theoretical guarantee for SyncRank](https://mlsanity.com/paper/2509.22766/) (2025)
- [From Authors to Reviewers: Leveraging Rankings to Improve Peer Review](https://mlsanity.com/paper/2510.21726/) (2025)
- [Bradley-Terry model under general comparison graphs](https://mlsanity.com/paper/2610.06237/) (2026)

## Abstract (arXiv)

This paper studies the performance of the spectral method in the estimation and uncertainty quantification of the unobserved preference scores of compared entities in a general and more realistic setup. Specifically, the comparison graph consists of hyper-edges of possible heterogeneous sizes, and the number of comparisons can be as low as one for a given hyper-edge. Such a setting is pervasive in real applications, circumventing the need to specify the graph randomness and the restrictive homogeneous sampling assumption imposed in the commonly used Bradley-Terry-Luce (BTL) or Plackett-Luce (PL) models. Furthermore, in scenarios where the BTL or PL models are appropriate, we unravel the relationship between the spectral estimator and the Maximum Likelihood Estimator (MLE). We discover that a two-step spectral method, where we apply the optimal weighting estimated from the equal weighting vanilla spectral method, can achieve the same asymptotic efficiency as the MLE. Given the asymptotic distributions of the estimated preference scores, we also introduce a comprehensive framework to carry out both one-sample and two-sample ranking inferences, applicable to both fixed and random graph settings. It is noteworthy that this is the first time effective two-sample rank testing methods have been proposed. Finally, we substantiate our findings via comprehensive numerical simulations and subsequently apply our developed methodologies to perform statistical inferences for statistical journals and movie rankings.