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Though the statistical analysis of ranking data has been a subject ofinterest over the past centuries, especially in economics, psychology or socialchoice theory, it has been revitalized in the past 15 years by recentapplications such as recommender or search engines and is receiving nowincreasing interest in the machine learning literature. Numerous modern systemsindeed generate ranking data, representing for instance ordered results to aquery or user preferences. Each such ranking usually involves a small butvarying subset of the whole catalog of items only. The study of the variabilityof these data, i.e. the statistical analysis of incomplete rank-ings, ishowever a great statistical and computational challenge, because of theirheterogeneity and the related combinatorial complexity of the problem. Whereasmany statistical methods for analyzing full rankings (orderings of all theitems in the catalog) are documented in the dedicated literature, partialrankings (full rankings with ties) or pairwise comparisons, only a fewapproaches are available today to deal with incomplete ranking, relying each ona strong specific assumption. It is the purpose of this article to introduce anovel general framework for the statistical analysis of incomplete rankings. Itis based on a representation tailored to these specific data, whoseconstruction is also explained here, which fits with the natural multi-scalestructure of incomplete rankings and provides a new decomposition of rankinformation with a multiresolu-tion analysis interpretation (MRA). We show thatthe MRA representation naturally allows to overcome both the statistical andcomputational challenges without any structural assumption on the data. Ittherefore provides a general and flexible framework to solve a wide variety ofstatistical problems, where data are of the form of incomplete rankings.

A Multiresolution Analysis Framework for the Statistical Analysis of Incomplete Rankings