Multiresolution Analysis of Incomplete Rankings

Stéphan Clémençon (1) , Jérémie Jakubowicz (2, 3) , Eric Sibony (1)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 SAMOVAR - Services répartis, Architectures, MOdélisation, Validation, Administration des Réseaux
3 CNRS - Centre National de la Recherche Scientifique

Incomplete rankings on a set of items {1, ..., n} are orderings of the form a_{1} \prec ... \prec a_{k}, with {a_{1}, ..., a_{k}} \subset {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.

Mots clés: Incomplete rankings; Multiresolution Analysis; Wavelets; Injective Words

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Stéphan Clémençon, Jérémie Jakubowicz, Eric Sibony. Multiresolution Analysis of Incomplete Rankings. 2014. ⟨hal-00957087⟩