# Deep Matrix Factorization Approach for Collaborative Filtering Recommender Systems
## 3 Results
This normalization seems to be crucial for the bad performance of FAuST as ss: this restriction might be very useful for image processing and memory saving problems which are the usual applications of FAuST, but it seems to be very ill-posed for regression problems. Due to the failure of this method as ss, it was decided to pull apart the existing deep factorization methods, designed specifically for image processing, and to choose the competing baselines from the most popular mf-based cf methods, namely PMF [29], NMF [30] and SVD++ [31]. These baselines contain several hyper-parameters that must be tuned in order to improve their performance on the selected datasets. The optimal configuration of the hyper-parameters of each baseline has been found by a grid search optimization. All combinations resulting from evaluating a wide range of values for each hyper-parameter have been evaluated, and the combination returning \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dataset** & **Number of Users** & **Number of Items** & **Number of Ratings** & **Number of Test Ratings** & **Possible Scores** \\ \hline MovieLens 100K & 943 & 1682 & 92,026 & 7974 & 1 to 5 stars \\ MovieLens 1M & 6040 & 3706 & 911,031 & 89,178 & 1 to 5 stars \\ FilmTrust & 1508 & 2071 & 32,675 & 2819 & 0.5 to 4.0 with half increments \\ MyAnimeList & 69,600 & 9927 & 5,788,207 & 549,027 & 1 to 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the datasets used in the experiments. the least mae on the predictions of the test split has been selected as the most suitable for each baseline and data set. Table 2 contains the resulting hyper-parameters of this optimization process. Recall that \(k\) denotes the number of hidden factors in the factorization, \(\gamma\) is the learning rate of the associated gradient descent optimization and \(\lambda\) is the regularization hyper-parameter. Analogously, DeepMF also contains several hyper-parameters needed to fit the model to an specific dataset. As with the baselines, we performed a grid search optimization in order to obtain the combination of parameters that minimizes the prediction error (mae) in the test split. In this case, the recursive nature of the proposed method causes that the number of hyper-parameter combinations to be evaluated grows exponentially. To deal with this combinatorial explosion, we evaluated with depths from 1 to 4, i.e., performing 1 to 4 deep factorizations. Each of this factorizations has been evaluated varying the number of latent factors \(k\in\{3,6,9\}\), the learning rate \(\gamma\in\{0.01,0.1\}\) and the regularization \(\lambda\in\{0.01,0.1\}\). The number of iterations has been fixed to \(T=[50,50,\ldots,50]\) for all factorizations. In total, 22,620 combinations of hyper-parameters has been evaluated for each dataset. Table 3 contains the top five results returned by the grid search optimization. Note that hyper-parameters has been named as in Algorithm 1 to facilitate their interpretation. Once all the evaluated models have been set up, the best tuned models can be compared to measure the prediction and recommendation improvement of DeepMF with respect to the selected baselines. Table 4 contains the mae of the predictions performed by all the evaluated models. We can observe that the proposed model DeepMF significantly improves the accuracy of predictions with respect to other models in MovieLens 100K, MovieLens 1M and FilmTrust datasets. Similarly, in the MyAnimeList dataset, DeepMF substantially improves PMF and NMF baselines and achieves a slightly