An Adaptive Matrix Factorization Approach for Personalized Recommender Systems

Given a set $U$ of users and a set of items $I$, a dataset of recommendations can be viewed as a sparse rectangular matrix $A$ of size $|U|\times |I|$ such that $a_{u,i}$ contains the rating the user $u$ assigns to item $i$, $a_{u,i}=?$ if the user $u$ has not rated the item $i$. The goal of a recommender system is to predict replacements to the missing observations $?$ in $A$ in order to make personalized recommendations meeting the user's tastes. A promising approach is the one based on the low-rank nonnegative matrix factorization of $A$ where items and users are represented in terms of a few vectors. These vector can be used for estimating the missing evaluations and to produce new recommendations. In this paper we propose an algorithm based on the nonnegative matrix Factorization approach for predicting the missing entries. Numerical test have been performed to estimate the accuracy in predicting the missing entries and in the recommendations provided and we have compared our technique with others in the literature. We have tested the algorithm on the MovieLens databases containing ratings of users on movies.