G. Biau

Titre : Statistical Analysis of $k$-Nearest Neighbor Collaborative 
Recommendation

Abstract :

 Collaborative recommendation is an information-filtering technique that 
attempts to present information items (movies, music, books, news, images, 
Web pages, etc.) that are likely of interest to the Internet user. 
Traditionally, collaborative systems deal with situations with two types 
of variables, users and items. In its most common form, the problem is 
framed as trying to estimate ratings for items that have not yet been 
consumed by a user. Despite wide-ranging literature, little is known about 
the statistical properties of recommendation systems. In fact, no clear 
probabilistic model even exists allowing us to precisely describe the 
mathematical forces driving collaborative filtering. To provide an initial 
contribution to this, we propose to set out a general sequential stochastic 
model for collaborative recommendation and analyze its asymptotic performance 
as the number of users grows.  We offer an in-depth analysis of the so-called 
cosine-type nearest neighbor collaborative method, which is one of the most 
widely used algorithms in collaborative filtering. We establish consistency 
of the procedure under mild assumptions on the model. Rates of convergence 
and examples are also provided.

This is a joint work with Benoît Cadre (ENS Cachan, Antenne de Rennes) 
and Laurent Rouvière (Université Rennes 2 and ENSAI).