# Antithetic and Monte Carlo kernel estimators for partial rankings

arXiv: https://arxiv.org/abs/1807.00400 · HTML: https://mlsanity.com/paper/1807.00400/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2018.

## Built on by (most cited first, 11 of 11)

- [Sampling Permutations for Shapley Value Estimation](https://mlsanity.com/paper/2104.12199/) (2021)
- [The Shapley Value in Machine Learning](https://mlsanity.com/paper/2202.05594/) (2022)
- [Approximating the Shapley Value without Marginal Contributions](https://mlsanity.com/paper/2302.00736/) (2023)
- [CoRTX: Contrastive Framework for Real-time Explanation](https://mlsanity.com/paper/2303.02794/) (2023)
- [Efficient XAI Techniques: A Taxonomic Survey](https://mlsanity.com/paper/2302.03225/) (2023)
- [Accelerating Shapley Explanation via Contributive Cooperator Selection](https://mlsanity.com/paper/2206.08529/) (2022)
- [BaCO: A Fast and Portable Bayesian Compiler Optimization Framework](https://mlsanity.com/paper/2212.11142/) (2022)
- [Efficient and Accurate Learning of Mixtures of Plackett-Luce Models](https://mlsanity.com/paper/2302.05343/) (2023)
- [Shift-Invariant Attribute Scoring for Kolmogorov-Arnold Networks via Shapley Value](https://mlsanity.com/paper/2510.01663/) (2025)
- [The Graph Cut Kernel for Ranked Data](https://mlsanity.com/paper/2105.12356/) (2021)
- [Kernel-based External Validity of Experimental Studies](https://mlsanity.com/paper/2610.13625/) (2026)

## Abstract (arXiv)

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 prevents 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.