Multiresolution analysis of incomplete rankings with applications to prediction
Résumé
Data representing preferences of users are a typical example of the Big Datasets modern technologies, such as e- commerce portals, now permit to collect, in an explicit or implicit fashion. Such data are highly complex, insofar as the number of items n for which users may possibly express their preferences is explosive and the collection of items or products a given user actually examines and is capable of comparing is highly variable and of extremely low cardinality compared to n. It is the main purpose of this paper to promote a new representation of preference data, viewed as incomplete rankings. In contrast to alternative approaches, the very nature of preference data is preserved by the "multiscale analysis" we propose, identifying here "scale" with the set of items over which preferences are expressed, whose construction relies on recent results in algebraic topology. The representation of preference data it provides shares similarities with wavelet multiresolution analysis on a Euclidean space and can be computed at a reasonable cost given the complexity of the original data. Beyond computational and theoretical advantages, the "wavelet like" transform is shown to compress preference data into relatively few basis coefficients and thus facilitates statistical tasks such as distribution estimation or prediction. This is illustrated here by very encouraging empirical work based on popular benchmark real datasets