Matrix factorization (MF) is a powerful approach used in recommender systems. One main drawback of MF is the difculty to interpret the automatically formed features. Following the intuition that the relation between users and items can be expressed through a reduced set of users, referred to as representative users, we propose a simple modi cation of a traditional MF algorithm, that forms a set of features corresponding to these representative users. On one state of the art dataset, we show that the proposed representative users-based non-negative matrix factorization (RU-NMF) discovers interpretable features, while slightly (in some cases insigni cantly) decreasing the accuracy.
trix R into two low-rank matrices U (dim(U ) = K M )
and V (dim(V ) = K N ) in such a way that the product
of these matrices approximates the original rating matrix
R R = U T V . The set of K factors can be seen as a joint
latent space on which a mapping of both users and items
spaces is performed [
Step 1. A traditional matrix factorization is performed.
Following [
Step 2. A normalization of each of the M column vectors of the matrix U is performed so as to result in unitary columns. The resulting normalized matrix is denoted by Unorm and the vector of normalization coe cients by C.
Step 3. This step is dedicated to the identi cation of the representative users in the Unorm matrix. A user um is considered as the best preimage candidate (representative user) for the feature fk if the vector unmorm is the closest to the corresponding canonical vector (a vector with the only one non-zero and equal to 1 value on the position k). The notion of closeness between vectors is expressed in Euclidean distance. Once all representative users are identi ed, the matrix Unorm is modi ed so as to obtain a matrix in a form of (1): lines, corresponding to the representative users, are replaced with appropriate canonic vectors. The resulting modi ed matrix is denoted by Unmoordm.
Step 4. Each column of the matrix Unmoordm is multiplied by the appropriate normalization coe cient from the set C resulting in matrix U mod. After this, representative users will remain preimages of the features but with scaling factors. Step 5. In order to obtain the best model we also have to modify the matrix V . The modi cation of V can be performed using optimization methods with the starting value obtained during the rst step. As the objective of this paper is to determine the relevance of nding preimages of the features and to quantify the decrease in the quality of the recommendations, we did not consider this step. Step 6. The resulting recommendation model is made up of matrices U mod and V (R = U mod T V ). 3. EXPERIMENTAL RESULTS Experiments are performed on the 100k MovieLens dataset1, with 80% of ratings used for learning the model and 20% for testing it. The accuracy is evaluated with two classical measures: mean absolute error (MAE) and root mean square error (RMSE). The goal of the experiments is to compare the accuracies of RU-NMF and NMF. For these reasons we compute the accuracy loss = err(RU-eNrMr(FN)MFer)r(NMF) 100% for factorizations with 10, 15 and 20 features on 30 di erent samples. Results are presented in Table 1. A positive loss means that NMF performs better than RU-NMF. In the worst case the accuracy loss equals to 6.64%, for RMSE with 20 features, which is quite small. The lowest average accuracy loss (0:05%) is obtained with 10 features for both errors. When comparing the accuracy loss between test and learning sets, we can note that the average loss is 3 times 1http://grouplens.org/datasets/movielens/ 10 features mean min max 15 features mean min max 20 features mean min max lower on test than on learning, for both errors and for all the number of features: thus we can say that RU-NMF has a lower relative loss between learn and test compared to NMF. A thorough analysis of the losses obtained on the 30 samples has shown that the accuracy loss on the test set is lower than the one on the learning set in all cases. In some runs, RU-NMF has even a higher accuracy than NMF (Table 1, values in gray shadow). 4. DISCUSSIONS AND FUTURE WORK The analysis of the accuracy loss between RU-NMF and traditional NMF has shown that prediction error rises slightly (in some cases insigni cantly) with RU-NMF. However the features formed with this approach consistently disturb the accuracy on the test set less than on the learning one. This can be considered as a potential ability of factorization techniques with features related to reality to form better searched predictions. The proposed approach also lets us easily explain the resulting recommendations. Indeed, each user of the population is linearly mapped on the basis related to representative users (through matrix U ) and the preferences of the latter ones (expressed by matrix V ) are used to estimate the ratings of the whole population. In a future work, we would like to focus rst of all on the veri cations of the hypothesis that users associated with the features can be really considered as representative ones. We believe that this can be done while solving the new item cold-start problem with ratings of the representative users on new items used to estimate ratings of all the population on these items.