# The Weighted Kendall and High-order Kernels for Permutations

arXiv: https://arxiv.org/abs/1802.08526 · HTML: https://mlsanity.com/paper/1802.08526/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Feb 2018.

## Built on by (most cited first, 2 of 2)

- [Bayesian Optimization over Permutation Spaces](https://mlsanity.com/paper/2112.01049/) (2021)
- [The Graph Cut Kernel for Ranked Data](https://mlsanity.com/paper/2105.12356/) (2021)

## Abstract (arXiv)

We propose new positive definite kernels for permutations. First we introduce a weighted version of the Kendall kernel, which allows to weight unequally the contributions of different item pairs in the permutations depending on their ranks. Like the Kendall kernel, we show that the weighted version is invariant to relabeling of items and can be computed efficiently in O(n ln(n)) operations, where n is the number of items in the permutation. Second, we propose a supervised approach to learn the weights by jointly optimizing them with the function estimated by a kernel machine. Third, while the Kendall kernel considers pairwise comparison between items, we extend it by considering higher-order comparisons among tuples of items and show that the supervised approach of learning the weights can be systematically generalized to higher-order permutation kernels.