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

• Activity Number: 347 - Recent Advances in Clustering and Mixture Models Analysis
• Type: Topic-Contributed
• Date/Time: Thursday, August 12, 2021 : 10:00 AM to 11:50 AM
• Sponsor: Section for Statistical Programmers and Analysts
• Author(s): Cheng Mao* and Yihong Wu
• Companies: Georgia Institute of Technology and Yale
• Keywords: mixture of permutations; Mallows model; rank aggregation; group of pairwise comparisons; method of moments

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.