Laurent Rouvière

Statistical analysis of k-nearest neighbor collaborative recommendation

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.

Parameter selection in modified histogram estimates

A multivariate modified histogram density estimate depending on a reference density g and a partition P has been proved to have good consistency properties according to several information theoretic criteria. Given an i.i.d. sample, we show how to select automatically both g and P so that the expected L 1 error of the corresponding selected estimate is within a given constant multiple of the best possible error plus an additive term which tends to zero under mild assumptions. Our method is inspired by the combinatorial tools developed by Devroye and Lugosi [Devroye, L. and Lugosi, G., 2001, Combinatorial Methods in Density Estimation (New York, NY: Springer–Verlag)] and it includes a wide range of reference density and partition models. Results of simulations are also presented.

Effective Construction of Modified Histograms in Higher Dimensions

Density estimation raises delicate problems in higher dimensions especially when strong convergence is required and data marginals can be highly correlated. Modified histograms have been introduced to circumvent the problem of low bin counts when convergence is considered in the sense of information divergence. These estimates are defined from some reference probability density and an associated partition which is defined in the univariate case fromni the quantiles of the reference density. Therefore, in the multivariate case, the definition of the partition causes an additional probleni related to the lack of total order. In this paper, we present a method for constructing modified multivariate histograms such that the corresponding partition is well adapted to the observed data. The approach is based on a data-driven coordinate system selected by cross-validation. We discuss the performance of our estimate with the help of a finite sample sirnulation study.