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

TL;DR

This paper introduces a polynomial-time algorithm for learning high-dimensional Mallows mixture models of permutations with optimal sample complexity, improving efficiency and understanding in rank aggregation applications.

Contribution

The work presents the first polynomial-time algorithm with optimal sample complexity for high-dimensional Mallows mixtures, extending noiseless demixing results to noisy models.

Findings

Algorithm achieves sample complexity proportional to log n

Characterizes sample complexity dependence on noise parameter

Extends noiseless demixing results to noisy Mallows models

Abstract

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.