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In applications such as rank aggregation, mixture models for permutations arefrequently used when the population exhibits heterogeneity. In this work, westudy the widely used Mallows mixture model. In the high-dimensional setting,we propose a polynomial-time algorithm that learns a Mallows mixture ofpermutations on $n$ elements with the optimal sample complexity that isproportional to $\log n$, improving upon previous results that scalepolynomially with $n$. In the high-noise regime, we characterize the optimaldependency of the sample complexity on the noise parameter. Both objectives areaccomplished by first studying demixing permutations under a noiseless querymodel using groups of pairwise comparisons, which can be viewed as moments ofthe mixing distribution, and then extending these results to the noisy Mallowsmodel by simulating the noiseless oracle.

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