The Weighted Kendall and High-order Kernels for Permutations

TL;DR

This paper introduces weighted and high-order kernels for permutations, enabling more flexible and efficient analysis of permutation data with learned weights and higher-order comparisons.

Contribution

It presents a novel weighted Kendall kernel, an efficient computation method, and a supervised approach to learn weights, extending to higher-order permutation kernels.

Findings

Weighted Kendall kernel is invariant to relabeling.

Kernel can be computed in $O(n \, \ln(n))$ time.

Supervised learning of weights improves permutation analysis.

Abstract

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.