Concentric mixtures of Mallows models for top-$k$ rankings: sampling and identifiability

TL;DR

This paper studies mixtures of concentric Mallows models for top-$k$ rankings, proposing algorithms for sampling, parameter learning, and separating components, with applications to noisy rank aggregation involving expert and non-expert voters.

Contribution

It introduces efficient algorithms for sampling, identifiability, and learning parameters of concentric Mallows mixtures, and adapts rank aggregation to handle noisy voter populations.

Findings

Proposed polynomial-time algorithms for component separation.

Bounded sample complexity for the Borda algorithm with top-$k$ rankings.

Demonstrated the ability to recover ground truth consensus rankings amidst noise.

Abstract

In this paper, we consider mixtures of two Mallows models for top-$k$ rankings, both with the same location parameter but with different scale parameters, i.e., a mixture of concentric Mallows models. This situation arises when we have a heterogeneous population of voters formed by two homogeneous populations, one of which is a subpopulation of expert voters while the other includes the non-expert voters. We propose efficient sampling algorithms for Mallows top-$k$ rankings. We show the identifiability of both components, and the learnability of their respective parameters in this setting by, first, bounding the sample complexity for the Borda algorithm with top-$k$ rankings and second, proposing polynomial time algorithm for the separation of the rankings in each component. Finally, since the rank aggregation will suffer from a large amount of noise introduced by the non-expert voters, we adapt the Borda algorithm to be able to recover the ground truth consensus ranking which is especially consistent with the expert rankings.