Antithetic and Monte Carlo kernel estimators for partial rankings

Noname manuscript No. (will be inserted by the editor) Abstract In the modern age, rankings data is ubiquitous and it is useful for a variety of applications such as recommender systems, multi-object tracking and preference learning. However, most rankings data encountered in the real world is incomplete, which forbids the direct application of existing modelling tools for complete rankings. Our contribution is a novel way to extend kernel methods for complete rankings to partial rankings, via consistent Monte Carlo estimators of Gram matrices. These Monte Carlo kernel estimators are based on extending kernel mean embeddings to the embedding of a set of full rankings consistent with an observed partial ranking. They form a computationally tractable alternative to previous approaches for partial rankings data. We also present a novel variance reduction scheme based on an antithetic variate construction between permutations to obtain an improved estimator. An overview of the existing kernels and metrics for permutations is also provided. Keywords Reproducing Kernel Hilbert Space; Partial rankings; Monte Carlo; Antithetic variates; Gram matrix 1 Motivation Permutations play a fundamental role in statistical modelling and machine learning applications involving rank-M.