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The Nonnegative Matrix Factorization (NMF) of the rating matrix has shown tobe an effective method to tackle the recommendation problem. In this paper wepropose new methods based on the NMF of the rating matrix and we compare themwith some classical algorithms such as the SVD and the regularized andunregularized non-negative matrix factorization approach. In particular a newalgorithm is obtained changing adaptively the function to be minimized at eachstep, realizing a sort of dynamic prior strategy. Another algorithm is obtainedmodifying the function to be minimized in the NMF formulation by enforcing thereconstruction of the unknown ratings toward a prior term. We then combinedifferent methods obtaining two mixed strategies which turn out to be veryeffective in the reconstruction of missing observations. We perform athoughtful comparison of different methods on the basis of several evaluationmeasures. We consider in particular rating, classification and ranking measuresshowing that the algorithm obtaining the best score for a given measure is ingeneral the best also when different measures are considered, lowering theinterest in designing specific evaluation measures. The algorithms have beentested on different datasets, in particular the 1M, and 10M MovieLens datasetscontaining ratings on movies, the Jester dataset with ranting on jokes andAmazon Fine Foods dataset with ratings on foods. The comparison of thedifferent algorithms, shows the good performance of methods employing both anexplicit and an implicit regularization scheme. Moreover we can get a boost bymixed strategies combining a fast method with a more accurate one.

Adaptive Nonnegative Matrix Factorization and Measure Comparisons for Recommender Systems