Abstract

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 rep-resenting 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 frame-work 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.