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The kendall and mallows kernels for permutations
 In Proceedings of the 32nd International Conference on Machine Learning (ICML15
, 1935
"... Abstract We show that the widely used Kendall tau correlation coefficient, and the related Mallows kernel, are positive definite kernels for permutations. They offer computationally attractive alternatives to more complex kernels on the symmetric group to learn from rankings, or to learn to rank. W ..."
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Abstract We show that the widely used Kendall tau correlation coefficient, and the related Mallows kernel, are positive definite kernels for permutations. They offer computationally attractive alternatives to more complex kernels on the symmetric group to learn from rankings, or to learn to rank. We show how to extend the Kendall kernel to partial rankings or rankings with uncertainty, and demonstrate promising results on highdimensional classification problems in biomedical applications.
DOI: 10.1007/9783642237805_32 Clustering Rankings in the Fourier Domain
, 2012
"... Abstract. It is the purpose of this paper to introduce a novel approach to clustering rank data on a set of possibly large cardinality n ∈ N ∗, relying upon Fourier representation of functions defined on the symmetric group Sn. In the present setup, covering a wide variety of practical situations, r ..."
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Abstract. It is the purpose of this paper to introduce a novel approach to clustering rank data on a set of possibly large cardinality n ∈ N ∗, relying upon Fourier representation of functions defined on the symmetric group Sn. In the present setup, covering a wide variety of practical situations, rank data are viewed as distributions on Sn. Cluster analysis aims at segmenting data into homogeneous subgroups, hopefully very dissimilar in a certain sense. Whereas considering dissimilarity measures/distances between distributions on the non commutative group Sn, in a coordinate manner by viewing it as embedded in the set [0, 1] n! for instance, hardly yields interpretable results and leads to face obvious computational issues, evaluating the closeness of groups of permutations in the Fourier domain may be much easier in contrast. Indeed, in a wide variety of situations, a few wellchosen Fourier (matrix) coefficients may permit to approximate efficiently two distributions on Sn as well as their degree of dissimilarity, while describing global properties in an interpretable fashion. Following in the footsteps of recent advances in automatic feature selection in the context of unsupervised learning, we propose to cast the task of clustering rankings in terms of optimization of a criterion that can be expressed in the Fourier domain in a simple manner. The effectiveness of the method proposed is illustrated by numerical experiments based on artificial and real data.
Multiresolution Analysis of Incomplete Rankings ∗
"... Incomplete rankings on a set of items {1,..., n} are orderings of the form a1 ≺ · · · ≺ ak, with {a1,... ak} ⊂ {1,..., n} and k < n. Though they arise in many modern applications, only a few methods have been introduced to manipulate them, most of them consisting in representing any incompl ..."
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Incomplete rankings on a set of items {1,..., n} are orderings of the form a1 ≺ · · · ≺ ak, with {a1,... ak} ⊂ {1,..., n} and k < n. Though they arise in many modern applications, only a few methods have been introduced to manipulate them, most of them consisting in representing any incomplete ranking by the set of all its possible linear extensions on {1,..., n}. It is the major purpose of this paper to introduce a completely novel approach, which allows to treat incomplete rankings directly, representing them as injective words over {1,..., n}. Unexpectedly, operations on incomplete rankings have very simple equivalents in this setting and the topological structure of the complex of injective words can be interpretated in a simple fashion from the perspective of ranking. We exploit this connection here and use recent results from algebraic topology to construct a multiresolution analysis and develop a wavelet framework for incomplete rankings. Though purely combinatorial, this construction relies on the same ideas underlying multiresolution analysis on a Euclidean space, and permits to localize the information related to rankings on each subset of items. It can be viewed as a crucial step toward nonlinear approximation of distributions of incomplete rankings and paves the way for many statistical applications, including preference data analysis and the design of recommender systems.