Title: The Weighted Kendall and High-order Kernels for Permutations,
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.