# Deep Matrix Factorization Approach for Collaborative Filtering Recommender Systems
## 1 Introduction
[17] provide an innovative DL architecture to improve CF results by exploiting the potential of the reliability concept to raise predictions and recommendations quality by incorporating prediction errors in the DL layers. Regarding convolutional neural networks, [PERSON] et al. [18] propose to use them in order to enhance CF by using the outer product instead of the customary dot product to model the interaction patterns of both users and items, as well as capturing the high-order correlations among embedding dimensions. In addition, [PERSON] and [PERSON] [19] proposed an artificial neural network that consists of a convolutional autoencoder along with a fully connected layer to learn robust deep features for classification. Along the same lines, Recurrent Neural networks have been used with RS[20; 21; 22], although under some restrictions (i.e., session-based recommendations), due to the features of this kind of deep architecture, which are suitable for sequential data processing. Even Generative Adversarial Networks (GANs) have been used to generate negative samples for a memory network-based streaming RS[23] and enhancing the Bayesian personalized ranking with adversarial training [24]. Another promising line of application of DL is to pull apart the neural networks and to apply the underlying philosophy to extend the usual MF algorithm to get deep factorizations. This approach is a common trend in several machine learning contexts, but it has never been applied to RS. For instance, [PERSON] and [PERSON] proposed FAuST [25], a deep factorization method of a matrix into a product of many sparse factors that can be achieved through hierarchical refinements, in the spirit of deep networks. FAuST has proven to be very successful in image processing, data compression and inverse linear problems but, as we will see, it fails to provide a competitive alternative when applied to RS. In a similar vein, [PERSON] et al. [26] devised a semi-supervised learning method based on Non-negative Matrix Factorization, which was applied to the problem of face clustering, while [PERSON] and [PERSON] [27] used a similar approach to perform dimension reduction and pattern recognition. A multilevel decomposition method used to derive a feature representation for speech recognition is presented in [28], which outperforms existing features for various speech recognition tasks. In this paper, we propose a novel method for incorporating DL to CF-based RS. For that purpose, we present a new CF method that combines the DL paradigm with MF in order to improve the quality of predictions and recommendations. The approach presented here is based on the principles of DL, but unlike state-of-the-art works, the proposed method does not use deep neural network architectures. Instead, it transforms the mf process into a layered process, in which each layer factors out the errors made in the previous layer. Roughly speaking, the proposed model works as follows. Consider a matrix \(R\) that collects the known ratings of the users of the rs to the supervised items. Typically, the mf looks for a factorization of the form \(R\approx P\cdot Q\) for some matrices \(P,Q\) of low rank. In this way, the matrix \(E^{1}=R-P\cdot Q\) measures the error attained with the mf. The key point is that, if we were able to exactly predict the error, then we could correct the output of our model by tuning it to take into account the expected error. To get this, our proposal is to deepen this factorization and to look for a factorization of the error \(E^{1}\approx P^{1}\cdot Q^{1}\), which gives rise to a'second order error' \(E^{2}=E^{1}-P^{1}\cdot Q^{1}\). This refinement of the expected error can be continued as many times as desired in order to achieve a better control of the error. In this way, as byproduct of the application of the dl process, we end up with a more accurate model that auto-corrects its predictions with the estimation of the expected error in the guess.