A Stochastic Model for 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.

💡 Research Summary

The paper tackles a fundamental gap in the literature on collaborative recommendation: the lack of a rigorous probabilistic framework that can describe the statistical forces behind modern recommender systems. Rather than treating the user‑item rating matrix as a static object, the authors introduce a sequential stochastic model in which users arrive over time. Each user (U_i) is associated with a latent feature vector (\mathbf{X}_i \in \mathbb{R}^d), and each item (V_j) with a fixed latent vector (\mathbf{Y}_j). The observed rating is modeled as a noisy inner product, \