Sorting from Noisy Information

TL;DR

This paper develops polynomial-time algorithms for inferring an unknown order from noisy comparison and permutation data, demonstrating that maximum likelihood solutions closely approximate the true order, with applications in ranking systems.

Contribution

It introduces efficient algorithms for maximum likelihood estimation in noisy ranking models, specifically for noisy comparisons and noisy order data, with proven proximity to the true permutation.

Findings

Algorithms solve ranking problems with high probability

Maximum likelihood solutions are close to true order

Applicable to sports and search ranking scenarios

Abstract

This paper studies problems of inferring order given noisy information. In these problems there is an unknown order (permutation) $\pi$ on $n$ elements denoted by $1,...,n$. We assume that information is generated in a way correlated with $\pi$. The goal is to find a maximum likelihood $\pi^*$ given the information observed. We will consider two different types of observations: noisy comparisons and noisy orders. The data in Noisy orders are permutations given from an exponential distribution correlated with \pi (this is also called the Mallow's model). The data in Noisy Comparisons is a signal given for each pair of elements which is correlated with their true ordering. In this paper we present polynomial time algorithms for solving both problems with high probability. As part of our proof we show that for both models the maximum likelihood solution $\pi^{\ast}$ is close to the original permutation $\pi$. Our results are of interest in applications to ranking, such as ranking in sports, or ranking of search items based on comparisons by experts.