The Weighted Kendall and High-order Kernels for Permutations
This work addresses the need for more flexible and learnable similarity measures in permutation-based machine learning tasks, such as ranking or preference modeling, but it is incremental as it builds upon existing Kendall kernel methods.
The authors tackled the problem of defining positive definite kernels for permutations by introducing a weighted Kendall kernel that allows unequal weighting of item pairs based on ranks, and extended it to higher-order comparisons among tuples of items, with computational efficiency maintained at O(n ln(n)) operations. They also proposed a supervised approach to learn these weights jointly with kernel machine functions, generalizing it systematically to higher-order kernels.
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.