Deep Matrix Factorization Approach for Collaborative Filtering Recommender Systems

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732286508"> <meta name="title" content="Deep Matrix Factorization Approach for Collaborative Filtering Recommender Systems"> <meta name="description" content="Providing useful information to the users by recommending highly demanded products and services is a fundamental part of the business of many top tier companies. Recommender Systems make use of many sources of information to provide users with accurate predictions and novel recommendations of items. Here we propose, DeepMF, a novel collaborative filtering method that combines the Deep Learning paradigm with Matrix Factorization (MF) to improve the quality of both predictions and recommendations made to the user. Specifically, DeepMF performs successive refinements of a MF model with a layered architecture that uses the acquired knowledge in a layer as input for subsequent layers. Experimental results showed that the quality of both the predictions and recommendations of DeepMF overcome the baselines." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/applsci-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="Deep Matrix Factorization Approach for Collaborative Filtering Recommender Systems"> <meta name="dc.creator" content="Raúl Lara-Cabrera"> <meta name="dc.creator" content="Ángel González-Prieto"> <meta name="dc.creator" content="Fernando Ortega"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Applied Sciences 2020, Vol. 10, Page 4926"> <meta name="dc.date" content="2020-07-17"> <meta name ="dc.identifier" content="10.3390/app10144926"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="Providing useful information to the users by recommending highly demanded products and services is a fundamental part of the business of many top tier companies. Recommender Systems make use of many sources of information to provide users with accurate predictions and novel recommendations of items. Here we propose, DeepMF, a novel collaborative filtering method that combines the Deep Learning paradigm with Matrix Factorization (MF) to improve the quality of both predictions and recommendations made to the user. Specifically, DeepMF performs successive refinements of a MF model with a layered architecture that uses the acquired knowledge in a layer as input for subsequent layers. Experimental results showed that the quality of both the predictions and recommendations of DeepMF overcome the baselines." > <meta name="dc.subject" content="deep learning" > <meta name="dc.subject" content="recommender systems" > <meta name="dc.subject" content="collaborative filtering" > <meta name="dc.subject" content="matrix factorization" > <meta name ="prism.issn" content="2076-3417"> <meta name ="prism.publicationName" content="Applied Sciences"> <meta name ="prism.publicationDate" content="2020-07-17"> <meta name ="prism.volume" content="10"> <meta name ="prism.number" content="14"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="4926" > <meta name="citation_issn" content="2076-3417"> <meta name="citation_journal_title" content="Applied Sciences"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="Deep Matrix Factorization Approach for Collaborative Filtering Recommender Systems"> <meta name="citation_publication_date" content="2020/1"> <meta name="citation_online_date" content="2020/07/17"> <meta name="citation_volume" content="10"> <meta name="citation_issue" content="14"> <meta name="citation_firstpage" content="4926"> <meta name="citation_author" content="Lara-Cabrera, Raúl"> <meta name="citation_author" content="González-Prieto, Ángel"> <meta name="citation_author" content="Ortega, Fernando"> <meta name="citation_doi" content="10.3390/app10144926"> <meta name="citation_id" content="mdpi-app10144926"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2076-3417/10/14/4926"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2076-3417/10/14/4926/pdf?version=1595218236"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2076-3417/10/14/4926/pdf?version=1595218236"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2076-3417/10/14/4926/pdf?version=1595218236"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2076-3417/10/14/4926/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2076-3417/10/14/4926/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2076-3417/10/14/4926/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2076-3417/10/14/4926/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2076-3417/10/14/4926/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2076-3417/10/14/4926/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/applsci-logo-social.png?8600e93ff98dbf14" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2076-3417/10/14/4926" /> <meta property="og:title" content="Deep Matrix Factorization Approach for Collaborative Filtering Recommender Systems" /> <meta property="og:description" content="Providing useful information to the users by recommending highly demanded products and services is a fundamental part of the business of many top tier companies. Recommender Systems make use of many sources of information to provide users with accurate predictions and novel recommendations of items. Here we propose, DeepMF, a novel collaborative filtering method that combines the Deep Learning paradigm with Matrix Factorization (MF) to improve the quality of both predictions and recommendations made to the user. Specifically, DeepMF performs successive refinements of a MF model with a layered architecture that uses the acquired knowledge in a layer as input for subsequent layers. 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class='profile-card-drop' data-dropdown='profile-card-drop3163567' data-options='is_hover:true, hover_timeout:5000'> Raúl Lara-Cabrera</div><div id="profile-card-drop3163567" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/bundles/mdpisciprofileslink/img/unknown-user.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Raúl Lara-Cabrera</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/876038?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.net/scholars?q=Ra%C3%BAl%20Lara-Cabrera" class="button button--color-inversed" 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!important; height: auto; top: -5px;"></a>, </span><span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop3163568' data-options='is_hover:true, hover_timeout:5000'> Ángel González-Prieto</div><div id="profile-card-drop3163568" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/profiles/888315/thumb/Ángel_González-Prieto.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Ángel González-Prieto</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/888315?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a 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class="affiliation-item"><sup>†</sup></div> <div class="affiliation-name ">These authors contributed equally to this work.</div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Appl. Sci.</em> <b>2020</b>, <em>10</em>(14), 4926; <a href="https://doi.org/10.3390/app10144926">https://doi.org/10.3390/app10144926</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 18 June 2020</span> / <span style="display: inline-block">Revised: 6 July 2020</span> / <span style="display: inline-block">Accepted: 15 July 2020</span> / <span style="display: inline-block">Published: 17 July 2020</span> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/applsci/special_issues/recommender_systems_collaborative_filtering ">Recommender Systems and Collaborative Filtering</a>)<br/> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-385965" aria-controls="drop-supplementary-385965" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-385965" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2076-3417/10/14/4926/pdf?version=1595218236" data-name="Deep Matrix Factorization Approach for Collaborative Filtering Recommender Systems" data-journal="applsci">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2076-3417/10/14/4926/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2076-3417/10/14/4926/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2076-3417/10/14/4926/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2076-3417/10/14/4926/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g001.png?1602245608" title=" <strong>Figure 1</strong><br/> <p>Illustrative explanation of proposed DeepMF method.</p> "> </a> <a href="https://pub.mdpi-res.com/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g002.png?1602245608" title=" <strong>Figure 2</strong><br/> <p>Quality of the recommendations measured by precision and recall. The higher the better. Blue number over the lines represents the size of the recommendation list for each value.</p> "> </a> <a href="https://pub.mdpi-res.com/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g003.png?1602245608" title=" <strong>Figure 3</strong><br/> <p>Average value of the predictions provided by each factorization according to its depth. Top 5 combinations of hyper-parameters for each dataset included in <a href="#applsci-10-04926-t003" class="html-table">Table 3</a> are shown.</p> "> </a> </div> <a class="button button--color-inversed" href="/2076-3417/10/14/4926/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">Providing useful information to the users by recommending highly demanded products and services is a fundamental part of the business of many top tier companies. Recommender Systems make use of many sources of information to provide users with accurate predictions and novel recommendations of items. Here we propose, DeepMF, a novel collaborative filtering method that combines the Deep Learning paradigm with Matrix Factorization (MF) to improve the quality of both predictions and recommendations made to the user. Specifically, DeepMF performs successive refinements of a MF model with a layered architecture that uses the acquired knowledge in a layer as input for subsequent layers. Experimental results showed that the quality of both the predictions and recommendations of DeepMF overcome the baselines.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=deep+learning">deep learning</a>; <a href="/search?q=recommender+systems">recommender systems</a>; <a href="/search?q=collaborative+filtering">collaborative filtering</a>; <a href="/search?q=matrix+factorization">matrix factorization</a></div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-applsci-10-04926' type='intro'><h2 data-nested='1'> 1. Introduction</h2><div class='html-p'>Presently, data processing has become a priority for the society. The amount of information we generate every day is on the rise, mainly due to the increase in the number of devices we use routinely as well as new patterns of interaction with technology that have been developed over the last few years. With such a high volume of real-time data, Machine Learning (<span class='html-small-caps'>ml</span>) techniques became a key tool for being able to extract knowledge from data. At present, <span class='html-small-caps'>ml</span> is one of the most active research field not only within computer science, but also in healthcare [<a href="#B1-applsci-10-04926" class="html-bibr">1</a>], sociology [<a href="#B2-applsci-10-04926" class="html-bibr">2</a>] and industry [<a href="#B3-applsci-10-04926" class="html-bibr">3</a>].</div><div class='html-p'>One of the most important applications of <span class='html-small-caps'>ml</span> are Recommender System (RS) [<a href="#B4-applsci-10-04926" class="html-bibr">4</a>], which are techniques that make use of different sources of information for providing users with predictions and recommendations of items, adjusting factors such as accuracy, novelty, sparsity and stability in the recommendations [<a href="#B5-applsci-10-04926" class="html-bibr">5</a>,<a href="#B6-applsci-10-04926" class="html-bibr">6</a>]. Top tier companies such as Netflix, Spotify and Amazon, use <span class='html-small-caps'>rs</span> provide useful information to the users by recommending highly demanded products and services. Although it is possible to find many kinds of <span class='html-small-caps'>rs</span> in the state of the art, they all share a key component: The filtering algorithm at the core. According to the filtering algorithm used, <span class='html-small-caps'>rs</span> might be categorized [<a href="#B7-applsci-10-04926" class="html-bibr">7</a>] into (a) Collaborative Filtering (CF), (b) demographic filtering, (c) content-based filtering and (d) hybrid filtering. Regarding <span class='html-small-caps'>cf</span>, Matrix Factorization (<span class='html-small-caps'>mf</span>) is one of its most widely used techniques [<a href="#B4-applsci-10-04926" class="html-bibr">4</a>,<a href="#B8-applsci-10-04926" class="html-bibr">8</a>]. It operates by decomposing the user-item interaction matrix into the product of two lower dimensional rectangular matrices as well as assigning different regularization weights to the latent factors in order to improve the prediction results [<a href="#B9-applsci-10-04926" class="html-bibr">9</a>].</div><div class='html-p'>A major recent breakthrough in the field of <span class='html-small-caps'>ml</span> is the so-called Deep Learning (<span class='html-small-caps'>dl</span>) technology, that has attracted much attention not only from the scientific community, but also from society in general. Oppositely to other <span class='html-small-caps'>ml</span> techniques, which can only extract knowledge from the shallow of the data, typically based on statistical evidence, <span class='html-small-caps'>dl</span> is able to detect and exploit the underlying multi-layer structure of the data. Although <span class='html-small-caps'>dl</span> is generally linked to artificial neural networks, the paradigm is more complex than that. As stated by LeCun et al. [<a href="#B10-applsci-10-04926" class="html-bibr">10</a>], “<span class='html-small-caps'>dl</span> allows computational models that are composed of multiple processing layers to learn representations of data with multiple levels of abstraction”. Since its inception, <span class='html-small-caps'>dl</span> has positioned as a very top notch technology for uncovering tangled structures in high-dimensional data. In this way, it has been applied to a wide range of domains such as speech recognition [<a href="#B11-applsci-10-04926" class="html-bibr">11</a>], image recognition [<a href="#B12-applsci-10-04926" class="html-bibr">12</a>] or natural language processing [<a href="#B13-applsci-10-04926" class="html-bibr">13</a>], among others.</div><div class='html-p'>As might be expected, <span class='html-small-caps'>rs</span> have not been immune to the boom in <span class='html-small-caps'>dl</span> and it is possible to find several works in this direction. The most obvious interplay is to apply deep neural networks [<a href="#B14-applsci-10-04926" class="html-bibr">14</a>] for improving the predictions of the RSs. Indeed, they provide appropriate a posteriori biases to the input data type, exploiting any inherent structure within the data. Furthermore, deep neural networks are capable of modeling the non-linearity in data with nonlinear activations, which is one of the principal aims of modern RS, making it possible to capture complex interaction patterns between users and items.</div><div class='html-p'>Due to the aforementioned advantages, neural networks with deep architectures, i.e., many layers, have been used to create new RS. For instance, DeepFM [<a href="#B15-applsci-10-04926" class="html-bibr">15</a>] is an approach that models the interactions of high and low order features through deep neural networks and factorization machines, respectively. On the other hand, Autoencoder-Based Collaborative Filtering (ACF) [<a href="#B16-applsci-10-04926" class="html-bibr">16</a>] uses another well-known <span class='html-small-caps'>dl</span> architecture, namely variational autoencoders, to build a <span class='html-small-caps'>rs</span> model that decompose the partial observed ratings vectors in a similar way as one-hot encoding. In the same way, Bobadilla et al. [<a href="#B17-applsci-10-04926" class="html-bibr">17</a>] provide an innovative <span class='html-small-caps'>dl</span> architecture to improve <span class='html-small-caps'>cf</span> results by exploiting the potential of the reliability concept to raise predictions and recommendations quality by incorporating prediction errors in the <span class='html-small-caps'>dl</span> layers. Regarding convolutional neural networks, He et al. [<a href="#B18-applsci-10-04926" class="html-bibr">18</a>] propose to use them in order to enhance <span class='html-small-caps'>cf</span> 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, Abavisani and Patel [<a href="#B19-applsci-10-04926" class="html-bibr">19</a>] 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 <span class='html-small-caps'>rs</span> [<a href="#B20-applsci-10-04926" class="html-bibr">20</a>,<a href="#B21-applsci-10-04926" class="html-bibr">21</a>,<a href="#B22-applsci-10-04926" class="html-bibr">22</a>], 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 <span class='html-small-caps'>rs</span> [<a href="#B23-applsci-10-04926" class="html-bibr">23</a>] and enhancing the Bayesian personalized ranking with adversarial training [<a href="#B24-applsci-10-04926" class="html-bibr">24</a>].</div><div class='html-p'>Another promising line of application of <span class='html-small-caps'>dl</span> is to pull apart the neural networks and to apply the underlying philosophy to extend the usual <span class='html-small-caps'>mf</span> 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, Le Magoarou and Gribonval proposed FAuST [<a href="#B25-applsci-10-04926" class="html-bibr">25</a>], 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, Trigeorgis et al. [<a href="#B26-applsci-10-04926" class="html-bibr">26</a>] devised a semi-supervised learning method based on Non-negative Matrix Factorization, which was applied to the problem of face clustering, while Guo and Zhang [<a href="#B27-applsci-10-04926" class="html-bibr">27</a>] 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 [<a href="#B28-applsci-10-04926" class="html-bibr">28</a>], which outperforms existing features for various speech recognition tasks.</div><div class='html-p'>In this paper, we propose a novel method for incorporating <span class='html-small-caps'>dl</span> to <span class='html-small-caps'>CF</span>-based RSs. For that purpose, we present a new <span class='html-small-caps'>cf</span> method that combines the <span class='html-small-caps'>dl</span> paradigm with <span class='html-small-caps'>mf</span> in order to improve the quality of predictions and recommendations. The approach presented here is based on the principles of <span class='html-small-caps'>dl</span>, but unlike state-of-the-art works, the proposed method does not use deep neural network architectures. Instead, it transforms the <span class='html-small-caps'>mf</span> process into a layered process, in which each layer factors out the errors made in the previous layer.</div><div class='html-p'>Roughly speaking, the proposed model works as follows. Consider a matrix <span class='html-italic'>R</span> that collects the known ratings of the users of the <span class='html-small-caps'>rs</span> to the supervised items. Typically, the <span class='html-small-caps'>mf</span> looks for a factorization of the form <math display='inline'><semantics> <mrow> <mi>R</mi> <mo>≈</mo> <mi>P</mi> <mo>·</mo> <mi>Q</mi> </mrow> </semantics></math> for some matrices <math display='inline'><semantics> <mrow> <mi>P</mi> <mo>,</mo> <mi>Q</mi> </mrow> </semantics></math> of low rank. In this way, the matrix <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mn>1</mn> </msup> <mo>=</mo> <mi>R</mi> <mo>−</mo> <mi>P</mi> <mo>·</mo> <mi>Q</mi> </mrow> </semantics></math> measures the error attained with the <span class='html-small-caps'>mf</span>. 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 <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mn>1</mn> </msup> <mo>≈</mo> <msup> <mi>P</mi> <mn>1</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>1</mn> </msup> </mrow> </semantics></math>, which gives rise to a ‘second order error’ <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>E</mi> <mn>1</mn> </msup> <mo>−</mo> <msup> <mi>P</mi> <mn>1</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>1</mn> </msup> </mrow> </semantics></math>. 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 <span class='html-small-caps'>dl</span> process, we end up with a more accurate model that auto-corrects its predictions with the estimation of the expected error in the guess.</div><div class='html-p'>Summarizing, the proposed <span class='html-small-caps'>rs</span> model follows the <span class='html-small-caps'>dl</span> paradigm by performing successive refinements of a <span class='html-small-caps'>mf</span> model with a layered architecture and letting it to use the acquired knowledge in one layer to be used as input for subsequent layers. The novelty of our work lies in using the concepts of <span class='html-small-caps'>dl</span>, not for use with artificial neural networks, but to improve the <span class='html-small-caps'>mf</span> itself.</div><div class='html-p'>The rest of the paper is structured as follows. <a href="#sec2-applsci-10-04926" class="html-sec">Section 2</a> contains the material and methods used to formalize the proposed method, as well as the associated algorithm. <a href="#sec3-applsci-10-04926" class="html-sec">Section 3</a> presents the results of the experimental evaluation conducted in order to measure the performance of the presented method. As we will see, the proposed algorithm consistently outperforms the previous <span class='html-small-caps'>mf</span> methods that can be found in the literature. Finally, <a href="#sec4-applsci-10-04926" class="html-sec">Section 4</a> discusses about these results and its impact on the <span class='html-small-caps'>rs</span> research field.</div></section><section id='sec2-applsci-10-04926' type=''><h2 data-nested='1'> 2. Materials and Methods</h2><div class='html-p'>A current trend in the field of <span class='html-small-caps'>cf</span> is to improve the quality of the predictions by means of different techniques of <span class='html-small-caps'>mf</span>, such as PMF [<a href="#B29-applsci-10-04926" class="html-bibr">29</a>], NMF [<a href="#B30-applsci-10-04926" class="html-bibr">30</a>], and SVD++ [<a href="#B31-applsci-10-04926" class="html-bibr">31</a>]. However, all these techniques rely on the same approach: they pose an optimization problem through a loss function that measures the divergence of the model of the expected behavior of users and items in a collaborative context from the actual behavior. Trying to break with this paradigm, in this paper we present a novel recommendation model that, using the <span class='html-small-caps'>mf</span> paradigm, introduces the principles of <span class='html-small-caps'>dl</span> to refine the output of the model through successive trainings. We named this new model Deep Matrix Factorization (DeepMF).</div><div class='html-p'><a href="#applsci-10-04926-f001" class="html-fig">Figure 1</a> summarizes the operation of DeepMF. As we can observe, the model is initialized with the usual input of a CF-based RS: a matrix <span class='html-italic'>R</span> that contains the ratings of the users to the items. Note that this matrix is sparse, since a user generally only votes a very small subset of the existing items. As in classical <span class='html-small-caps'>mf</span> approaches, this matrix <span class='html-italic'>R</span>, that will be also called <math display='inline'><semantics> <mrow> <mi>R</mi> <mo>=</mo> <msup> <mi>E</mi> <mn>0</mn> </msup> </mrow> </semantics></math> in our context, is approximated by a dense matrix <math display='inline'><semantics> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> </semantics></math> of the form <math display='inline'><semantics> <mrow> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> <mo>=</mo> <msup> <mi>P</mi> <mn>0</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>0</mn> </msup> </mrow> </semantics></math>, where <math display='inline'><semantics> <msup> <mi>P</mi> <mn>0</mn> </msup> </semantics></math> and <math display='inline'><semantics> <msup> <mi>Q</mi> <mn>0</mn> </msup> </semantics></math> are matrices of small rank. This matrix <math display='inline'><semantics> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> </semantics></math> provides the predicted ratings at the first step.</div><div class='html-p'>At this point the <span class='html-small-caps'>dl</span> begins. A new matrix <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mn>1</mn> </msup> <mo>=</mo> <mi>R</mi> <mo>−</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> </mrow> </semantics></math> is built by computing the attained errors between the original ratings <span class='html-italic'>R</span> and the predicted ratings stored in <math display='inline'><semantics> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> </semantics></math>. This new matrix is, again, approximated by a factorization into two new small rank matrices <math display='inline'><semantics> <mrow> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>1</mn> </msup> <mo>=</mo> <msup> <mi>P</mi> <mn>1</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>1</mn> </msup> </mrow> </semantics></math>, which produces the errors at the second step <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>E</mi> <mn>1</mn> </msup> <mo>−</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>1</mn> </msup> </mrow> </semantics></math>. This process is repeated as many times as desired, by generating and factorizing successive error matrices <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mn>1</mn> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>E</mi> <mi>N</mi> </msup> </mrow> </semantics></math>. Presumably, this sequence of error matrices converges to zero, so we get preciser predictions as we add new layers to the model.</div><div class='html-p'>This method can be formalized as follows. Suppose that our <span class='html-small-caps'>cf</span>-based <span class='html-small-caps'>rs</span> is dealing with <span class='html-italic'>n</span> users and <span class='html-italic'>m</span> items, whose ratings are collected in a spare matrix <math display='inline'><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mrow> <mi>n</mi> <mo>×</mo> <mi>m</mi> </mrow> </msup> </mrow> </semantics></math>, where <math display='inline'><semantics> <msub> <mi>R</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </semantics></math> is the rating that the user <span class='html-italic'>u</span> gave to item <span class='html-italic'>i</span> (typically, integer values between 1 and 5), or <math display='inline'><semantics> <mrow> <msub> <mi>R</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>•</mo> </mrow> </semantics></math> if <span class='html-italic'>u</span> has not rated the item <span class='html-italic'>i</span>. We look for a factorization of <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mn>0</mn> </msup> <mo>=</mo> <mi>R</mi> </mrow> </semantics></math> of the form <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mn>0</mn> </msup> <mo>≈</mo> <msup> <mi>P</mi> <mn>0</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>0</mn> </msup> </mrow> </semantics></math>, where <math display='inline'><semantics> <msup> <mi>P</mi> <mn>0</mn> </msup> </semantics></math> is a <math display='inline'><semantics> <mrow> <mi>n</mi> <mo>×</mo> <msup> <mi>k</mi> <mn>0</mn> </msup> </mrow> </semantics></math> matrix and <math display='inline'><semantics> <msup> <mi>Q</mi> <mn>0</mn> </msup> </semantics></math> is a <math display='inline'><semantics> <mrow> <msup> <mi>k</mi> <mn>0</mn> </msup> <mo>×</mo> <mi>m</mi> </mrow> </semantics></math> matrix. Here, <math display='inline'><semantics> <msup> <mi>k</mi> <mn>0</mn> </msup> </semantics></math> is interpreted as the number of hidden factors we try to detect in the first step (typically <math display='inline'><semantics> <msup> <mi>k</mi> <mn>0</mn> </msup> </semantics></math> is around 10) and <math display='inline'><semantics> <mrow> <msup> <mi>Q</mi> <mn>0</mn> </msup> <mo>,</mo> <msup> <mi>P</mi> <mn>0</mn> </msup> </mrow> </semantics></math> are seen as the projection/co-projection of the <span class='html-italic'>n</span> users and <span class='html-italic'>m</span> items into a <math display='inline'><semantics> <msup> <mi>k</mi> <mn>0</mn> </msup> </semantics></math>-dimensional latent space. These small rank matrices are learned such that the product <math display='inline'><semantics> <mrow> <msup> <mi>P</mi> <mn>0</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>0</mn> </msup> </mrow> </semantics></math> is a good approximation of the ratings matrix <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mn>0</mn> </msup> <mo>=</mo> <mi>R</mi> </mrow> </semantics></math>, that is <div class='html-disp-formula-info' id='FD1-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mi>E</mi> <mn>0</mn> </msup> <mo>≈</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> <mo>=</mo> <msup> <mi>P</mi> <mn>0</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>0</mn> </msup> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> in the usual euclidean distance.</div><div class='html-p'>By subtracting the approximation performed by <math display='inline'><semantics> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> </semantics></math> to the original matrix <span class='html-italic'>R</span> we can obtain a new sparse matrix <math display='inline'><semantics> <msup> <mi>E</mi> <mn>1</mn> </msup> </semantics></math> that contains the prediction error <div class='html-disp-formula-info' id='FD2-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mi>E</mi> <mn>1</mn> </msup> <mo>=</mo> <mi>R</mi> <mo>−</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> <mo>=</mo> <msup> <mi>E</mi> <mn>0</mn> </msup> <mo>−</mo> <msup> <mi>P</mi> <mn>0</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>0</mn> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div></div><div class='html-p'>Note that positive values in <math display='inline'><semantics> <msup> <mi>E</mi> <mn>1</mn> </msup> </semantics></math> denotes that the prediction is too low and must be increased and negative values in <math display='inline'><semantics> <msub> <mi>E</mi> <mn>1</mn> </msub> </semantics></math> denotes that the prediction is too high and must be decreased. As we stated before, the main contribution of the proposed method is its deep learning approach. This is achieved by performing a new factorization to the <math display='inline'><semantics> <msup> <mi>E</mi> <mn>1</mn> </msup> </semantics></math> error matrix in such a way that <div class='html-disp-formula-info' id='FD3-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mi>E</mi> <mn>1</mn> </msup> <mo>≈</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>1</mn> </msup> <mo>=</mo> <msup> <mi>P</mi> <mn>1</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>1</mn> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div></div><div class='html-p'>The matrices <math display='inline'><semantics> <mrow> <msup> <mi>P</mi> <mn>1</mn> </msup> <mo>,</mo> <msup> <mi>Q</mi> <mn>1</mn> </msup> </mrow> </semantics></math> have orders <math display='inline'><semantics> <mrow> <mi>n</mi> <mo>×</mo> <msup> <mi>k</mi> <mn>1</mn> </msup> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msup> <mi>k</mi> <mn>1</mn> </msup> <mo>×</mo> <mi>m</mi> </mrow> </semantics></math> for a certain number of latent factors <math display='inline'><semantics> <msup> <mi>k</mi> <mn>1</mn> </msup> </semantics></math>. Observe that we may take <math display='inline'><semantics> <mrow> <msup> <mi>k</mi> <mn>1</mn> </msup> <mo>≠</mo> <msup> <mi>k</mi> <mn>0</mn> </msup> </mrow> </semantics></math> in order to get a different level of resolution in the factorization.</div><div class='html-p'>In the general case, if we computed <math display='inline'><semantics> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> steps of the deep learning procedure, we compute the <span class='html-italic'>s</span>-th matrix of errors as <div class='html-disp-formula-info' id='FD4-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mi>E</mi> <mi>s</mi> </msup> <mo>=</mo> <msup> <mi>E</mi> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>E</mi> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>P</mi> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div> and we look for a factorization into matrices <math display='inline'><semantics> <msup> <mi>P</mi> <mi>s</mi> </msup> </semantics></math> of order <math display='inline'><semantics> <mrow> <mi>n</mi> <mo>×</mo> <msup> <mi>k</mi> <mi>s</mi> </msup> </mrow> </semantics></math> and <math display='inline'><semantics> <msup> <mi>Q</mi> <mi>s</mi> </msup> </semantics></math> of order <math display='inline'><semantics> <mrow> <msup> <mi>k</mi> <mi>s</mi> </msup> <mo>×</mo> <mi>m</mi> </mrow> </semantics></math> such that, as much as possible in the euclidean norm, <div class='html-disp-formula-info' id='FD5-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mi>E</mi> <mi>s</mi> </msup> <mo>≈</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mi>s</mi> </msup> <mo>=</mo> <msup> <mi>P</mi> <mi>s</mi> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mi>s</mi> </msup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div></div><div class='html-p'>Once the deep factorization process ends after <span class='html-italic'>N</span> steps, the original rating matrix <span class='html-italic'>R</span> can be reconstructed by adding the estimates of the errors as <div class='html-disp-formula-info' id='FD6-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <mi>R</mi> <mo>≈</mo> <mover accent="true"> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> <mo>+</mo> <msup> <mi>E</mi> <mn>1</mn> </msup> <mo>=</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> <mo>+</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>1</mn> </msup> <mo>+</mo> <msup> <mi>E</mi> <mn>2</mn> </msup> <mo>=</mo> <mo>…</mo> <mo>≈</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>0</mn> </msup> <mo>+</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>1</mn> </msup> <mo>+</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mn>2</mn> </msup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">^</mo> </mover> <mi>N</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mspace width="1.em"/> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <msup> <mi>P</mi> <mn>0</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>0</mn> </msup> <mo>+</mo> <msup> <mi>P</mi> <mn>1</mn> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mn>1</mn> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>P</mi> <mi>N</mi> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mi>N</mi> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>N</mi> </munderover> <msup> <mi>P</mi> <mi>s</mi> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mi>s</mi> </msup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div></div><div class='html-p'>For any step <math display='inline'><semantics> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mrow> </semantics></math>, the factorization <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mi>s</mi> </msup> <mo>≈</mo> <msup> <mi>P</mi> <mi>s</mi> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mi>s</mi> </msup> </mrow> </semantics></math> is sought by the standard method of minimizing the euclidean distance between <math display='inline'><semantics> <msup> <mi>E</mi> <mi>s</mi> </msup> </semantics></math> and <math display='inline'><semantics> <mrow> <msup> <mi>P</mi> <mi>s</mi> </msup> <msup> <mi>Q</mi> <mi>s</mi> </msup> </mrow> </semantics></math> by gradient descent with regularization, as in PMF [<a href="#B29-applsci-10-04926" class="html-bibr">29</a>]. In this way, if we write <math display='inline'><semantics> <mrow> <msup> <mi>E</mi> <mi>s</mi> </msup> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>e</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> <mi>s</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, the rows of <math display='inline'><semantics> <msup> <mi>P</mi> <mi>s</mi> </msup> </semantics></math> are denoted by <math display='inline'><semantics> <mrow> <msubsup> <mi>p</mi> <mn>1</mn> <mi>s</mi> </msubsup> <mo>,</mo> <msubsup> <mi>p</mi> <mn>2</mn> <mi>s</mi> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>p</mi> <mi>n</mi> <mi>s</mi> </msubsup> </mrow> </semantics></math> and the columns of <math display='inline'><semantics> <msup> <mi>Q</mi> <mi>s</mi> </msup> </semantics></math> are denoted by <math display='inline'><semantics> <mrow> <msubsup> <mi>q</mi> <mn>1</mn> <mi>s</mi> </msubsup> <mo>,</mo> <msubsup> <mi>q</mi> <mn>2</mn> <mi>s</mi> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>q</mi> <mi>m</mi> <mi>s</mi> </msubsup> </mrow> </semantics></math>, the loss function is given by <div class='html-disp-formula-info' id='FD7-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi mathvariant="script">F</mi> <mi>s</mi> </msup> <mrow> <mo>(</mo> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> <mo>,</mo> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <msup> <mi>E</mi> <mi>s</mi> </msup> <mo>−</mo> <msup> <mi>P</mi> <mi>s</mi> </msup> <mo>·</mo> <msup> <mi>Q</mi> <mi>s</mi> </msup> <msup> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>λ</mi> <mi>s</mi> </msup> <mfenced separators="" open="(" close=")"> <munderover> <mo>∑</mo> <mrow> <mi>u</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> <msup> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> <msup> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mspace width="1.em"/> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow> <msubsup> <mi>e</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> <mi>s</mi> </msubsup> <mo>≠</mo> <mo>•</mo> </mrow> </munder> <msup> <mfenced separators="" open="(" close=")"> <msubsup> <mi>e</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> <mi>s</mi> </msubsup> <mo>−</mo> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> <mo>·</mo> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> </mfenced> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>λ</mi> <mi>s</mi> </msup> <mfenced separators="" open="(" close=")"> <munderover> <mo>∑</mo> <mrow> <mi>u</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> <msup> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> <msup> <mrow> <mo>|</mo> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div> where <math display='inline'><semantics> <msup> <mi>λ</mi> <mi>s</mi> </msup> </semantics></math> is the regularization hyper-parameter of the <span class='html-italic'>s</span>-th step to avoid overfitting.</div><div class='html-p'>The previous loss function <math display='inline'><semantics> <mi mathvariant="script">F</mi> </semantics></math> can be derived with respect to <math display='inline'><semantics> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> </semantics></math> and <math display='inline'><semantics> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> </semantics></math> resulting the following update rules for learning the model parameters <div class='html-disp-formula-info' id='FD8-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mtable displaystyle="true"> <mtr> <mtd columnalign="right"> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> </mtd> <mtd columnalign="left"> <mrow> <mo>←</mo> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> <mo>+</mo> <msup> <mi>γ</mi> <mi>s</mi> </msup> <mfenced separators="" open="(" close=")"> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>e</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> <mi>s</mi> </msubsup> <mo>−</mo> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> <mo>·</mo> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>·</mo> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> <mo>−</mo> <msup> <mi>λ</mi> <mi>s</mi> </msup> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> </mtd> <mtd columnalign="left"> <mrow> <mo>←</mo> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> <mo>+</mo> <msup> <mi>γ</mi> <mi>s</mi> </msup> <mfenced separators="" open="(" close=")"> <mrow> <mo stretchy="false">>(</mo> <msubsup> <mi>e</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> <mi>s</mi> </msubsup> <mo>−</mo> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> <mo>·</mo> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> <mo stretchy="false">>)</mo> </mrow> <mo>·</mo> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> <mo>−</mo> <msup> <mi>λ</mi> <mi>s</mi> </msup> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div> where <math display='inline'><semantics> <msup> <mi>γ</mi> <mi>s</mi> </msup> </semantics></math> is the learning rate hyper-parameter of the <span class='html-italic'>s</span>-th step to control the learning speed.</div><div class='html-p'>In this way, after finishing the nested factorization, the predicted ratings are collected in the matrix <math display='inline'><semantics> <mrow> <mover accent="true"> <mi>R</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mrow> <mo stretchy="false">>(</mo> <msub> <mover accent="true"> <mi>R</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">>)</mo> </mrow> </mrow> </semantics></math>, where the predicted rating of the user <span class='html-italic'>u</span> to the item <span class='html-italic'>i</span> is given by <div class='html-disp-formula-info' id='FD9-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mover accent="true"> <mi>R</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>N</mi> </munderover> <msubsup> <mi>p</mi> <mi>u</mi> <mi>s</mi> </msubsup> <mo>·</mo> <msubsup> <mi>q</mi> <mi>i</mi> <mi>s</mi> </msubsup> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div></div><div class='html-p'>Note that the proposed method consists of successive repetitions of a <span class='html-small-caps'>mf</span> process using the results of the previous <span class='html-small-caps'>mf</span> as input. Therefore, it is possible to make an algorithmic implementation of the proposed method using a recursive approach. Algorithm 1 contains the pseudo-code of the recursive implementation of DeepMF. The algorithm receives as input the <span class='html-italic'>E</span> matrix, which contains either the users’ ratings to the items (<math display='inline'><semantics> <mrow> <mi>R</mi> <mo>=</mo> <msup> <mi>E</mi> <mn>0</mn> </msup> </mrow> </semantics></math>) for the first call or the errors of the previous factorization <math display='inline'><semantics> <msup> <mi>E</mi> <mi>s</mi> </msup> </semantics></math> for the successive ones. It also receives the model hyper-parameters: the number of latent factors on each step (<span class='html-italic'>K</span>), the number of iterations of the gradient descent optimization on each step (<span class='html-italic'>T</span>), the learning rates (<math display='inline'><semantics> <mo>Γ</mo> </semantics></math>) and the regularizations (<math display='inline'><semantics> <mo>Λ</mo> </semantics></math>).</div><div class='html-p'>Note that these hyper-parameters were stacked so that each of the factorizations performed uses different hyper-parameters. The hyper-parameters of the first factorization will be placed at the top of the stack, those of the second factorization in the next one and so on until the parameters of the deeper factorization, which will be placed at the bottom of the stack. This allows us to define the stopping criteria of the algorithm: as soon as a stack is empty the learning process will be finished. Similarly, the output of the algorithm will be a stack containing the pairs <math display='inline'><semantics> <mrow> <mo>⟨</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo>⟩</mo> </mrow> </semantics></math> with the hidden factors of each of the factorizations carried out.</div><table class='html-array_table'><tbody ><tr ><td align='left' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-left' ><b>Algorithm 1:</b> Pseudo-code of a recursive implementation of DeepMF.</td></tr><tr ><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><span class='html-fig-inline' id = applsci-10-04926-i001 > <img alt="Applsci 10 04926 i001" data-lsrc="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-i001.png" /></span></td></tr></tbody></table><div class='html-p'>Algorithm 1 needs to be run in a single thread since the update of the <span class='html-italic'>P</span> matrix requires the <span class='html-italic'>Q</span> matrix and vice versa. However, this algorithm can easily be sped-up by using an Alternating Least Squares (<span class='html-small-caps'>als</span>) approach. Algorithm 2 contains a parrallel implementation of DeepMF using an <span class='html-small-caps'>als</span> approach. As can be observed, to perform a factorization, the matrices <span class='html-italic'>P</span> and <span class='html-italic'>Q</span> are updated <span class='html-italic'>t</span> times according to the loss function gradient. This update is performed in two steps: first, the values of the <span class='html-italic'>Q</span> matrix are fixed as constants allowing updating of the latent factors of each user (<math display='inline'><semantics> <msub> <mi>P</mi> <mi>u</mi> </msub> </semantics></math>) independently from the latent factors of the rest of the users, thus, they can be updated in parallel for each user. Second, the values of the <span class='html-italic'>P</span> matrix are fixed as constants and the latent factors of each item <math display='inline'><semantics> <msub> <mi>Q</mi> <mi>i</mi> </msub> </semantics></math> are updated independently from the latent factors of the rest of the items, thus, they can be updated in parallel for each item. Although this parallel implementation needs to go through the rating set twice to update both the <span class='html-italic'>P</span> and the <span class='html-italic'>Q</span> matrices, its computation time is significantly less than that of Algorithm 1 when more than 2 execution threads are available. For example, in a dataset with 1 million ratings, the total computation time of each iteration will be the time required to update the Algorithm 2, the total computation time in a processor with 8 execution threads of each iteration will be the time required to update the matrices <span class='html-italic'>P</span> and <span class='html-italic'>Q</span> <span class='html-italic'>2 * 1 million / 8 threads = 250,000 times</span>, a quarter of the total computation time required by Algorithm 1.</div><table class='html-array_table'><tbody ><tr ><td align='left' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-left' ><b>Algorithm 2:</b> Pseudo-code of a parrallel recursive implementation of Deep<span class='html-small-caps'>mf</span> using an <span class='html-small-caps'>als</span> approach.</td></tr><tr ><td align='left' valign='middle' style='border-bottom:solid thin' class='html-align-left' ><span class='html-fig-inline' id = applsci-10-04926-i002 > <img alt="Applsci 10 04926 i002" data-lsrc="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-i002.png" /></span></td></tr></tbody></table></section><section id='sec3-applsci-10-04926' type='results'><h2 data-nested='1'> 3. Results</h2><div class='html-p'>In this section we describe the empirical experiments designed to evaluate the performance of the proposed method. According to the standard <span class='html-small-caps'>cf</span>’s evaluation framework [<a href="#B32-applsci-10-04926" class="html-bibr">32</a>], the evaluation of the quality of the predictions is carried out by measuring the Mean Absolute Error (<span class='html-small-caps'>mae</span>) of the model output. The <span class='html-small-caps'>mae</span> is defined as average absolute difference between the actual ratings <span class='html-italic'>R</span> and the predicted ratings <math display='inline'><semantics> <mover accent="true"> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> of the test split <math display='inline'><semantics> <msup> <mi>R</mi> <mi>test</mi> </msup> </semantics></math>:<div class='html-disp-formula-info' id='FD10-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mrow> <mi>MAE</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mo>#</mo> <msup> <mi>R</mi> <mi>test</mi> </msup> </mrow> </mfrac> <munder> <mo>∑</mo> <mrow> <mrow> <mo>⟨</mo> <mi>u</mi> <mo>,</mo> <mi>i</mi> <mo>⟩</mo> </mrow> <mo>∈</mo> <msup> <mi>R</mi> <mi>test</mi> </msup> </mrow> </munder> <mrow> <mo>|</mo> <msub> <mi>R</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mover accent="true"> <mi>R</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>|</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div></div><div class='html-p'>Analogously, the quality of the top <span class='html-italic'>l</span> recommendations is evaluated by the precision and recall quality measures on the recommendation lists. Remark that, given an user <span class='html-italic'>u</span>, his recommendation list of <span class='html-italic'>l</span> items, <math display='inline'><semantics> <msubsup> <mi>L</mi> <mi>u</mi> <mi>l</mi> </msubsup> </semantics></math>, is the collection of the <span class='html-italic'>l</span> items <math display='inline'><semantics> <mrow> <msub> <mi>i</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>i</mi> <mi>l</mi> </msub> </mrow> </semantics></math> with <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>R</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>u</mi> <mo>,</mo> <msub> <mi>i</mi> <mn>1</mn> </msub> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mover accent="true"> <mi>R</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>u</mi> <mo>,</mo> <msub> <mi>i</mi> <mi>l</mi> </msub> </mrow> </msub> </mrow> </semantics></math> the highest <span class='html-italic'>l</span> predicted ratings.</div><div class='html-p'>Now, we fix a parameter <math display='inline'><semantics> <mrow> <mi>θ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </semantics></math>, which plays the role of a minimum threshold to discern whether or not an item is of interest to the users. The precision is defined as the average proportion of successful recommendations of the recommendation list:<div class='html-disp-formula-info' id='FD11-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mrow> <mi>precision</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mo>#</mo> <msup> <mi>U</mi> <mi>test</mi> </msup> </mrow> </mfrac> <munder> <mo>∑</mo> <mrow> <mi>u</mi> <mo>∈</mo> <msup> <mi>U</mi> <mi>test</mi> </msup> </mrow> </munder> <mfrac> <mfenced separators="" open="{" close="}"> <mi>i</mi> <mo>∈</mo> <msubsup> <mi>L</mi> <mi>u</mi> <mi>l</mi> </msubsup> <mrow> <mo>|</mo> </mrow> <msub> <mi>R</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mi>θ</mi> </mfenced> <mi>l</mi> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div> where <math display='inline'><semantics> <msup> <mi>U</mi> <mi>test</mi> </msup> </semantics></math> is the collection of users in the test split.</div><div class='html-p'>In the same vein, the recall is defined as the averaged proportion of successful recommendations included in the recommendation list with respect to the total number of test items the user <span class='html-italic'>u</span> likes:<div class='html-disp-formula-info' id='FD12-applsci-10-04926'> <div class='f'> <math display='block'><semantics> <mrow> <mi>recall</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mo>#</mo> <msup> <mi>U</mi> <mi>test</mi> </msup> </mrow> </mfrac> <munder> <mo>∑</mo> <mrow> <mi>u</mi> <mo>∈</mo> <msup> <mi>U</mi> <mi>test</mi> </msup> </mrow> </munder> <mfrac> <mfenced separators="" open="{" close="}"> <mi>i</mi> <mo>∈</mo> <msubsup> <mi>L</mi> <mi>u</mi> <mi>l</mi> </msubsup> <mrow> <mo>|</mo> </mrow> <msub> <mi>R</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mi>θ</mi> </mfenced> <mfenced separators="" open="{" close="}"> <mi>i</mi> <mo>∈</mo> <msubsup> <mi>R</mi> <mi>u</mi> <mi>test</mi> </msubsup> <mrow> <mo>|</mo> </mrow> <msub> <mi>R</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mi>θ</mi> </mfenced> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div> where <math display='inline'><semantics> <msubsup> <mi>R</mi> <mi>u</mi> <mi>test</mi> </msubsup> </semantics></math> is the collection of items in the test split at the <span class='html-italic'>u</span>-th row of <span class='html-italic'>R</span>.</div><div class='html-p'>The experimental evaluation has been carried out using MovieLens [<a href="#B33-applsci-10-04926" class="html-bibr">33</a>], FilmTrust [<a href="#B34-applsci-10-04926" class="html-bibr">34</a>] and MyAnimeList [<a href="#B35-applsci-10-04926" class="html-bibr">35</a>] datasets. <a href="#applsci-10-04926-t001" class="html-table">Table 1</a> contains their main parameters. To ensure the reproducibility of these experiments, all of them have been run using the benchmark version of these datasets included in Collaborative Filtering for Java (CF4J) [<a href="#B36-applsci-10-04926" class="html-bibr">36</a>].</div><div class='html-p'>The selection of the baselines has been done trying to pick a representative sample of the existing <span class='html-small-caps'>mf</span> models. A first approach, based on the deep nature of the proposed method, is to focus on existing deep factorization models such as FAuST [<a href="#B25-applsci-10-04926" class="html-bibr">25</a>] and DNMF [<a href="#B27-applsci-10-04926" class="html-bibr">27</a>]. However recall that, as stated in <a href="#sec1-applsci-10-04926" class="html-sec">Section 1</a>, these methods were designed for image processing, not for solving <span class='html-small-caps'>cf</span> problems, so a performance similar to other standard MF-based <span class='html-small-caps'>cf</span> algorithms is not guaranteed. Trying to discern whether these methods would be suitable for <span class='html-small-caps'>cf</span>, we proposed a preliminary experiment for evaluating the performance of FAuST against the MovieLens 100K dataset. To tune the hyper-parameters, a grid search was conducted whose limits are as follows. It was searched for factorizations of the rating matrix <span class='html-italic'>R</span> into up to 4 smaller matrices, with 6 or 9 hidden factors for each level and a gradient descent step size (learning rate) of <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.0001</mn> </mrow> </semantics></math> or <math display='inline'><semantics> <mrow> <mn>0.01</mn> </mrow> </semantics></math>. Notice that FAuST also allows you to impose a sparsity constraint on each factor matrix. For this experiment, it was allowed a <math display='inline'><semantics> <mrow> <mn>0</mn> <mo>%</mo> </mrow> </semantics></math> of sparsity (i.e., fully dense matrix with no restrictions) or a <math display='inline'><semantics> <mrow> <mn>20</mn> <mo>%</mo> </mrow> </semantics></math> of sparsity (that is, up to <math display='inline'><semantics> <mrow> <mn>80</mn> <mo>%</mo> </mrow> </semantics></math> of non-vanishing entries are allowed). The best found model attained a <span class='html-small-caps'>mae</span> of 3.49262 with a factorization into 4 matrices, all with 9 hidden factors, no sparsity constraints and a learning rate of <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>. This result is very far from the current baselines reported in the literature, which are typically around <math display='inline'><semantics> <mrow> <mn>0.75</mn> </mrow> </semantics></math> as we will see below, showing that the existing deep methods are unsuitable as RSs. In is worthy to mention that, although FAuST with no sparsity constraints is very similar to the usual PMF [<a href="#B29-applsci-10-04926" class="html-bibr">29</a>] algorithm, it looks for a factorization into normalized matrices with Frobenius norm 1. This normalization seems to be crucial for the bad performance of FAuST as RS: 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.</div><div class='html-p'>Due to the failure of this method as RS, 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 <span class='html-small-caps'>cf</span> methods, namely PMF [<a href="#B29-applsci-10-04926" class="html-bibr">29</a>], NMF [<a href="#B30-applsci-10-04926" class="html-bibr">30</a>] and SVD++ [<a href="#B31-applsci-10-04926" class="html-bibr">31</a>].</div><div class='html-p'>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 the least <span class='html-small-caps'>mae</span> on the predictions of the test split has been selected as the most suitable for each baseline and data set. <a href="#applsci-10-04926-t002" class="html-table">Table 2</a> contains the resulting hyper-parameters of this optimization process. Recall that <span class='html-italic'>k</span> denotes the number of hidden factors in the factorization, <math display='inline'><semantics> <mi>γ</mi> </semantics></math> is the learning rate of the associated gradient descent optimization and <math display='inline'><semantics> <mi>λ</mi> </semantics></math> is the regularization hyper-parameter.</div><div class='html-p'>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 (<span class='html-small-caps'>mae</span>) 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.</div><div class='html-p'>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 <math display='inline'><semantics> <mrow> <mi>k</mi> <mo>∈</mo> <mo>{</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>9</mn> <mo>}</mo> </mrow> </semantics></math>, the learning rate <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>∈</mo> <mo>{</mo> <mn>0</mn> <mo>.</mo> <mn>01</mn> <mo>,</mo> <mn>0.1</mn> <mo>}</mo> </mrow> </semantics></math> and the regularization <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>∈</mo> <mo>{</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.1</mn> <mo>}</mo> </mrow> </semantics></math>. The number of iterations has been fixed to <math display='inline'><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mo>[</mo> <mn>50</mn> <mo>,</mo> <mn>50</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>50</mn> <mo>]</mo> </mrow> </semantics></math> for all factorizations. In total, 22,620 combinations of hyper-parameters has been evaluated for each dataset. <a href="#applsci-10-04926-t003" class="html-table">Table 3</a> 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.</div><div class='html-p'>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. <a href="#applsci-10-04926-t004" class="html-table">Table 4</a> contains the <span class='html-small-caps'>mae</span> 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 worse <span class='html-small-caps'>mae</span> than SVD++. Observe that this later underperformance might be due to the fact that MyAnimeList is significantly larger than any other dataset, so deeper models of DeepMF would be needed in order the exploit the recursive nature of the proposed method that fit the dataset better than SVD++.</div><div class='html-p'><a href="#applsci-10-04926-f002" class="html-fig">Figure 2</a> contains the precision and recall of the recommendations when varying the number of desired top recommendations from <math display='inline'><semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> to <math display='inline'><semantics> <mrow> <mi>l</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> items. The threshold <math display='inline'><semantics> <mi>θ</mi> </semantics></math> for precision and recall has been set to <math display='inline'><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> for MovieLens datasets, <math display='inline'><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math> for FilmTrust dataset and <math display='inline'><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> for MyAnimeList dataset. We can observe that in all evaluated datasets the proposed method DeepMF provides the best balance between the precision and recall quality measures for any number of recommendations.</div><div class='html-p'>All experiments conducted in this article are committed to reproducible science. The full code of these experiments is available at <a href='https://github.com/ferortega/deep-matrix-factorization' target='_blank' rel="noopener noreferrer">https://github.com/ferortega/deep-matrix-factorization</a>.</div></section><section id='sec4-applsci-10-04926' type='discussion'><h2 data-nested='1'> 4. Discussion</h2><div class='html-p'>In this paper, we presented DeepMF, a novel MF-based <span class='html-small-caps'>cf</span> algorithm using a <span class='html-small-caps'>dl</span> approach. As stated in <a href="#sec1-applsci-10-04926" class="html-sec">Section 1</a> the <span class='html-small-caps'>dl</span> is an incipient approach in the <span class='html-small-caps'>ml</span> field that consists of a hierarchical self-training method that uses the information acquired from data. Although <span class='html-small-caps'>dl</span> has been used mainly in artificial neural networks, the proposed model breaks this tendency by applying the foundations of <span class='html-small-caps'>dl</span> to the field of MF.</div><div class='html-p'>The essence of the proposed method lies in the <span class='html-small-caps'>dl</span> approach. Our model performs successive matrix factorizations in order to successively refine the model output. Thus, the first factorization, the least deep one, represents the classic approach of <span class='html-small-caps'>mf</span>-based <span class='html-small-caps'>cf</span> models and seeks to predict the score that a user will give to an item. The second factorization seeks to refine the previous prediction by trying to increase the predictions that tend to be lower than the real rating and decrease those that tend to be higher. Subsequently, the learning is deepened by correcting the errors of the errors to build a <span class='html-small-caps'>dl</span> model that converges towards a prediction as close as possible to the real value to be learned.</div><div class='html-p'>This expected behavior is corroborated by the results of the experiment shown in <a href="#applsci-10-04926-f003" class="html-fig">Figure 3</a>. On it, we plot the average value of the predictions provided by each factorization layer according to its depth. The same decreasing trend in the average prediction is observed in all the analyzed datasets: as factorization is deeper, errors to be refined tend to zero and the learning process converges. The <a href="#applsci-10-04926-f003" class="html-fig">Figure 3</a> includes the top 5 combinations of hyper-parameters obtained in the grid search shown in <a href="#applsci-10-04926-t003" class="html-table">Table 3</a>. Note that a logarithmic scale has been used on the <span class='html-italic'>y</span>-axis to emphasize the differences in the deeper factorizations.</div><div class='html-p'>The hypothesis of this contribution was that a <span class='html-small-caps'>dl</span> approach applied to a <span class='html-small-caps'>mf</span>-based <span class='html-small-caps'>cf</span> can improve the quality of both predictions and recommendations. This hypothesis has been confirmed with the experimental results showed in <a href="#sec3-applsci-10-04926" class="html-sec">Section 3</a>. The quality of the predictions (see <a href="#applsci-10-04926-t004" class="html-table">Table 4</a>) and recommendations (see <a href="#applsci-10-04926-f002" class="html-fig">Figure 2</a>) of the proposed method, DeepMF, exceeds the baselines used in the 3 datasets analyzed: MovieLens 100K, MovieLens 1M and FilmTrust.</div><div class='html-p'>Summarizing, the proposed model in this paper expands the landscape of matrix factorization models by importing a Deep Learning approach from the field of neural computing. From this point, several future research lines open. Maybe, the most obvious one would be to analyze the <span class='html-small-caps'>dl</span> approach performed in this paper with other matrix factorization models and to define different loss functions depending on the depth of factorization. For example, experimental results show that SVD++ factorization works properly on MyAnimeList dataset, so it would be interesting to evaluate the performance of DeepMF using SVD++ as the initial factorization and other factorizations models for the deeper factorizations.</div><div class='html-p'>A more ambitious prospective work would be transferring the ideas of this paper to a purely bioinspired framework. For instance, it can be studied the incorporation of DeepMF as a model to be implemented by the neurons of a fully connected or convolutional neural network. In this way, the knowledge of the scientific community about network architectures can be applied to give rise to deeper an more involved nested patterns of matrix factorizations.</div></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>Conceptualization, R.L.-C. and Á.G.-P. and F.O.; formal analysis, Á.G.-P.; methodology, R.L.-C. and F.O.; software, A.G.-P. and F.O.; writing—original draft preparation, R.L.-C. and Á.G.-P. and F.O.; writing—review and editing, R.L.-C. All authors have read and agreed to the published version of the manuscript.</div></section><section class='html-notes'><h2 >Funding</h2><div class='html-p'>This work has been supported by Spanish Ministry of Science and Education and Competitivity (MINECO) and European Regional Development Fund (FEDER) under grants TIN2017-85727-C4-3-P (DeepBio).</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.</div></section><section id='html-references_list'><h2>References</h2><ol class='html-xx'><li id='B1-applsci-10-04926' class='html-x' data-content='1.'>Manogaran, G.; Lopez, D. A survey of big data architectures and machine learning algorithms in healthcare. <span class='html-italic'>Int. J. Biomed. Eng. 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[<a href="https://scholar.google.com/scholar_lookup?title=CF4J:+Collaborative+filtering+for+Java&author=Ortega,+F.&author=Zhu,+B.&author=Bobadilla,+J.&author=Hernando,+A.&publication_year=2018&journal=Knowl.+Based+Syst.&volume=152&pages=94%E2%80%9399&doi=10.1016/j.knosys.2018.04.008" class='google-scholar' target='_blank' rel='noopener noreferrer'>Google Scholar</a>] [<a href="https://doi.org/10.1016/j.knosys.2018.04.008" class='cross-ref' target='_blank' rel='noopener noreferrer'>CrossRef</a>]</li></ol></section><section id='FiguresandTables' type='display-objects'><div class="html-fig-wrap" id="applsci-10-04926-f001"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href="#fig_body_display_applsci-10-04926-f001"> <img alt="Applsci 10 04926 g001 550" data-large="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g001.png" data-original="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g001.png" data-lsrc="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g001-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href="#fig_body_display_applsci-10-04926-f001"></a> </div> </div> <div class="html-fig_description"> <b>Figure 1.</b> Illustrative explanation of proposed DeepMF method.  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_applsci-10-04926-f001" > <div class="html-caption" > <b>Figure 1.</b> Illustrative explanation of proposed DeepMF method.</div> <div class="html-img"><img alt="Applsci 10 04926 g001" data-large="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g001.png" data-original="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g001.png" data-lsrc="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g001.png" /></div> </div><div class="html-fig-wrap" id="applsci-10-04926-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href="#fig_body_display_applsci-10-04926-f002"> <img alt="Applsci 10 04926 g002 550" data-large="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g002.png" data-original="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g002.png" data-lsrc="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href="#fig_body_display_applsci-10-04926-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> Quality of the recommendations measured by precision and recall. The higher the better. Blue number over the lines represents the size of the recommendation list for each value.  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_applsci-10-04926-f002" > <div class="html-caption" > <b>Figure 2.</b> Quality of the recommendations measured by precision and recall. The higher the better. Blue number over the lines represents the size of the recommendation list for each value.</div> <div class="html-img"><img alt="Applsci 10 04926 g002" data-large="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g002.png" data-original="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g002.png" data-lsrc="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g002.png" /></div> </div><div class="html-fig-wrap" id="applsci-10-04926-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href="#fig_body_display_applsci-10-04926-f003"> <img alt="Applsci 10 04926 g003 550" data-large="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g003.png" data-original="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g003.png" data-lsrc="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href="#fig_body_display_applsci-10-04926-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> Average value of the predictions provided by each factorization according to its depth. Top 5 combinations of hyper-parameters for each dataset included in <a href="#applsci-10-04926-t003" class="html-table">Table 3</a> are shown.  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_applsci-10-04926-f003" > <div class="html-caption" > <b>Figure 3.</b> Average value of the predictions provided by each factorization according to its depth. Top 5 combinations of hyper-parameters for each dataset included in <a href="#applsci-10-04926-t003" class="html-table">Table 3</a> are shown.</div> <div class="html-img"><img alt="Applsci 10 04926 g003" data-large="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g003.png" data-original="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g003.png" data-lsrc="/applsci/applsci-10-04926/article_deploy/html/images/applsci-10-04926-g003.png" /></div> </div><div class="html-table-wrap" id="applsci-10-04926-t001"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href='#table_body_display_applsci-10-04926-t001'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href="#table_body_display_applsci-10-04926-t001"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 1.</b> Main parameters of the datasets used in the experiments. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_applsci-10-04926-t001" > <div class="html-caption" ><b>Table 1.</b> Main parameters of the datasets used in the experiments.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Dataset</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Number of Users</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Number of Items</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Number of Ratings</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Number of Test Ratings</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Possible Scores</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >MovieLens 100K</td><td align='center' valign='middle' class='html-align-center' >943</td><td align='center' valign='middle' class='html-align-center' >1682</td><td align='center' valign='middle' class='html-align-center' >92,026</td><td align='center' valign='middle' class='html-align-center' >7974</td><td align='center' valign='middle' class='html-align-center' >1 to 5 stars</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >MovieLens 1M</td><td align='center' valign='middle' class='html-align-center' >6040</td><td align='center' valign='middle' class='html-align-center' >3706</td><td align='center' valign='middle' class='html-align-center' >911,031</td><td align='center' valign='middle' class='html-align-center' >89,178</td><td align='center' valign='middle' class='html-align-center' >1 to 5 stars</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >FilmTrust</td><td align='center' valign='middle' class='html-align-center' >1508</td><td align='center' valign='middle' class='html-align-center' >2071</td><td align='center' valign='middle' class='html-align-center' >32,675</td><td align='center' valign='middle' class='html-align-center' >2819</td><td align='center' valign='middle' class='html-align-center' >0.5 to 4.0 with half increments</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >MyAnimeList</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >69,600</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >9927</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5,788,207</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >549,027</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1 to 10</td></tr></tbody> </table> </div><div class="html-table-wrap" id="applsci-10-04926-t002"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href='#table_body_display_applsci-10-04926-t002'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href="#table_body_display_applsci-10-04926-t002"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 2.</b> Best hyper-parameters for each baseline found by a grid search optimization. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_applsci-10-04926-t002" > <div class="html-caption" ><b>Table 2.</b> Best hyper-parameters for each baseline found by a grid search optimization.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Dataset</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >PMF</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >NMF</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >SVD++</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >MovieLens 100K</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.025</mn> </mrow> </semantics></math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.0014</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.08</mn> </mrow> </semantics></math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >MovieLens 1M</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.045</mn> </mrow> </semantics></math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.0014</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math></td></tr><tr ><td align='center' valign='middle' class='html-align-center' >FilmTrust</td><td align='center' valign='middle' class='html-align-center' ><math display='inline'><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.015</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' class='html-align-center' ><math display='inline'><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.0014</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >MyAnimeList</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.005</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.085</mn> </mrow> </semantics></math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.0014</mn> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.02</mn> </mrow> </semantics></math></td></tr></tbody> </table> </div><div class="html-table-wrap" id="applsci-10-04926-t003"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href='#table_body_display_applsci-10-04926-t003'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href="#table_body_display_applsci-10-04926-t003"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 3.</b> Top 5 results resulting from grid search optimization of DeepMF. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_applsci-10-04926-t003" > <div class="html-caption" ><b>Table 3.</b> Top 5 results resulting from grid search optimization of DeepMF.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Dataset</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Rank</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Hyper-Parameters</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >MAE</th></tr></thead><tbody ><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >MovieLens 100K</td><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [3, 3, 3, 3]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' class='html-align-center' >0.75017</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [3, 3, 3, 9]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' class='html-align-center' >0.75077</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [3, 6, 6, 3]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' class='html-align-center' >0.75079</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [3, 3, 3]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1]</td><td align='center' valign='middle' class='html-align-center' >0.75092</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><span class='html-italic'>K</span> = [3, 3, 9, 9]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.75105</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >MovieLens 1M</td><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [9, 3, 3]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.01, 0.01]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1]</td><td align='center' valign='middle' class='html-align-center' >0.70943</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [9, 3, 9, 3]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.01, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' class='html-align-center' >0.70948</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [9, 3, 9]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.01, 0.01]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1]</td><td align='center' valign='middle' class='html-align-center' >0.70949</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [9, 3, 3, 9]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.01, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' class='html-align-center' >0.70956</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><span class='html-italic'>K</span> = [9, 3, 3, 3]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.01, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.70959</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >FilmTrust</td><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [6, 6, 3]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.01]</td><td align='center' valign='middle' class='html-align-center' >0.64936</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [6, 6, 3]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1]</td><td align='center' valign='middle' class='html-align-center' >0.64987</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [6, 3, 6]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1]</td><td align='center' valign='middle' class='html-align-center' >0.65072</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [9, 3, 3]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1]</td><td align='center' valign='middle' class='html-align-center' >0.65088</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><span class='html-italic'>K</span> = [6, 6]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.01]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01]</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.65102</td></tr><tr ><td rowspan='5' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >MyAnimeList</td><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [9, 6, 9, 6]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' class='html-align-center' >0.97447</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [9, 9, 6, 9]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' class='html-align-center' >0.97452</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [9, 6, 6, 6]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' class='html-align-center' >0.97454</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' ><span class='html-italic'>K</span> = [9, 6, 6, 9]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' class='html-align-center' >0.97454</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><span class='html-italic'>K</span> = [9, 9, 9, 6]; <math display='inline'><semantics> <mrow> <mspace width="3.33333pt"/> <mo>Γ</mo> </mrow> </semantics></math>= [0.01, 0.1, 0.01, 0.1]; <math display='inline'><semantics> <mo>Λ</mo> </semantics></math> = [0.1, 0.01, 0.1, 0.01]</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.97458</td></tr></tbody> </table> </div><div class="html-table-wrap" id="applsci-10-04926-t004"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href='#table_body_display_applsci-10-04926-t004'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3417/10/14/4926/display" href="#table_body_display_applsci-10-04926-t004"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 4.</b> Quality of the predictions measured by the <span class='html-small-caps'>mae</span>. The lower the better. In bold the best recommendation model for each dataset. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_applsci-10-04926-t004" > <div class="html-caption" ><b>Table 4.</b> Quality of the predictions measured by the <span class='html-small-caps'>mae</span>. The lower the better. In bold the best recommendation model for each dataset.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >Dataset</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >DeepMF</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >PMF</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >NMF</th><th align='center' valign='middle' style='border-bottom:solid thin;border-top:solid thin' class='html-align-center' >SVD++</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >MovieLens 100K</td><td align='center' valign='middle' class='html-align-center' ><b>0.75017</b></td><td align='center' valign='middle' class='html-align-center' >0.76720</td><td align='center' valign='middle' class='html-align-center' >0.79138</td><td align='center' valign='middle' class='html-align-center' >0.78170</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >MovieLens 1M</td><td align='center' valign='middle' class='html-align-center' ><b>0.70943</b></td><td align='center' valign='middle' class='html-align-center' >0.71868</td><td align='center' valign='middle' class='html-align-center' >0.75166</td><td align='center' valign='middle' class='html-align-center' >0.74285</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >FilmTrust</td><td align='center' valign='middle' class='html-align-center' ><b>0.64936</b></td><td align='center' valign='middle' class='html-align-center' >0.84659</td><td align='center' valign='middle' class='html-align-center' >0.82911</td><td align='center' valign='middle' class='html-align-center' >0.65748</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >MyAnimeList</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.97447</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1.10006</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1.12025</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><b>0.95179</b></td></tr></tbody> </table> </div></section> <section id="html-copyright"><br>© 2020 by the authors. 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