The Weighted Kendall and High-order Kernels for Permutations

Yunlong Jiao, Jean-Philippe Vert
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:2314-2322, 2018.

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.

Cite this Paper


BibTeX
@InProceedings{pmlr-v80-jiao18a, title = {The Weighted {K}endall and High-order Kernels for Permutations}, author = {Jiao, Yunlong and Vert, Jean-Philippe}, booktitle = {Proceedings of the 35th International Conference on Machine Learning}, pages = {2314--2322}, year = {2018}, editor = {Dy, Jennifer and Krause, Andreas}, volume = {80}, series = {Proceedings of Machine Learning Research}, month = {10--15 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v80/jiao18a/jiao18a.pdf}, url = {http://proceedings.mlr.press/v80/jiao18a.html}, 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.} }
Endnote
%0 Conference Paper %T The Weighted Kendall and High-order Kernels for Permutations %A Yunlong Jiao %A Jean-Philippe Vert %B Proceedings of the 35th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2018 %E Jennifer Dy %E Andreas Krause %F pmlr-v80-jiao18a %I PMLR %P 2314--2322 %U http://proceedings.mlr.press/v80/jiao18a.html %V 80 %X 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.
APA
Jiao, Y. & Vert, J.. (2018). The Weighted Kendall and High-order Kernels for Permutations. Proceedings of the 35th International Conference on Machine Learning, in Proceedings of Machine Learning Research 80:2314-2322 Available from http://proceedings.mlr.press/v80/jiao18a.html.

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