  The Nonnegative Matrix Factorization (NMF) of the rating matrix has shown to
be an effective method to tackle the recommendation problem. In this paper we
propose new methods based on the NMF of the rating matrix and we compare them
with some classical algorithms such as the SVD and the regularized and
unregularized non-negative matrix factorization approach. In particular a new
algorithm is obtained changing adaptively the function to be minimized at each
step, realizing a sort of dynamic prior strategy. Another algorithm is obtained
modifying the function to be minimized in the NMF formulation by enforcing the
reconstruction of the unknown ratings toward a prior term. We then combine
different methods obtaining two mixed strategies which turn out to be very
effective in the reconstruction of missing observations. We perform a
thoughtful comparison of different methods on the basis of several evaluation
measures. We consider in particular rating, classification and ranking measures
showing that the algorithm obtaining the best score for a given measure is in
general the best also when different measures are considered, lowering the
interest in designing specific evaluation measures. The algorithms have been
tested on different datasets, in particular the 1M, and 10M MovieLens datasets
containing ratings on movies, the Jester dataset with ranting on jokes and
Amazon Fine Foods dataset with ratings on foods. The comparison of the
different algorithms, shows the good performance of methods employing both an
explicit and an implicit regularization scheme. Moreover we can get a boost by
mixed strategies combining a fast method with a more accurate one.
