On the identifiability of mixtures of ranking models

TL;DR

This paper proves that popular two-component mixture ranking models, including BTL and Plackett-Luce, are generically identifiable, meaning their parameters can be uniquely recovered except in rare cases, using algebraic geometry techniques.

Contribution

It establishes the generic identifiability of common mixture ranking models and introduces a general algebraic geometry framework for analyzing solution counts in polynomial systems.

Findings

Popular mixture models are generically identifiable.

A new algebraic geometry framework verifies solution counts.

The approach applies broadly to other learning models.

Abstract

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.