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Antithetic and Monte Carlo kernel estimators for partial rankings

Lomelí, M; Rowland, M; Gretton, A; Ghahramani, Z; (2019) Antithetic and Monte Carlo kernel estimators for partial rankings. Statistics and Computing , 29 pp. 1127-1147. 10.1007/s11222-019-09859-z. Green open access

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Abstract

In the modern age, rankings data are ubiquitous and they are 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 are incomplete, which prevent 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 for Gram matrices: matrices of kernel values between pairs of observations. We also present a novel variance-reduction scheme based on an antithetic variate construction between permutations to obtain an improved estimator for the Mallows kernel. The corresponding antithetic kernel estimator has lower variance, and we demonstrate empirically that it has a better performance in a variety of machine learning tasks. Both 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. An overview of the existing kernels and metrics for permutations is also provided.

Type: Article
Title: Antithetic and Monte Carlo kernel estimators for partial rankings
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s11222-019-09859-z
Publisher version: https://doi.org/10.1007/s11222-019-09859-z
Language: English
Additional information: Copyright © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Keywords: Reproducing kernel, Hilbert space, Partial rankings, Monte Carlo, Antithetic variates, Gram matrix
UCL classification: UCL
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Life Sciences
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Life Sciences > Gatsby Computational Neurosci Unit
URI: https://discovery.ucl.ac.uk/id/eprint/10076526
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