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Where <math>\|.\|_{\rm F}</math> is defined to be the [[frobenius norm]] whereas the other norms might be either frobenius or another norm depending on the specific recommending problem.<ref name="Paterek07">{{cite journal |last1=Paterek |first1=Arkadiusz |title=Improving regularized singular value decomposition for collaborative filtering |journal=Proceedings of KDD Cup and Workshop |date=2007 |url=https://www.mimuw.edu.pl/~paterek/ap_kdd.pdf}}</ref>
=== SVD++ ===
While Funk MF is able to provide very good recommendation quality, its ability to use only explicit numerical ratings as user-items interactions constitutes a limitation. Modern day [[recommender systems]] should exploit all available interactions both explicit (e.g. numerical ratings) and implicit (e.g. likes, purchases, skipped, bookmarked). To this end SVD++ was designed to take into account implicit interactions as well.<ref name="Cao15">{{cite book |last1=Cao |first1=Jian |last2=Hu |first2=Hengkui |last3=Luo |first3=Tianyan |last4=Wang |first4=Jia |last5=Huang |first5=May |last6=Wang |first6=Karl |last7=Wu |first7=Zhonghai |last8=Zhang |first8=Xing |title=Distributed Design and Implementation of SVD++ Algorithm for E-commerce Personalized Recommender System |volume=572 |date=2015 |pages=30–44 |doi=10.1007/978-981-10-0421-6_4 |publisher=Springer Singapore |language=en|series=Communications in Computer and Information Science |isbn=978-981-10-0420-9 }}</ref><ref name="Jia14">{{cite book |last1=Jia |first1=Yancheng |chapter=Users' brands preference based on SVD++ in recommender systems |journal=2014 IEEE Workshop on Advanced Research and Technology in Industry Applications (WARTIA) |pages=1175–1178 |date=September 2014 |doi=10.1109/wartia.2014.6976489 |publisher=IEEE|isbn=978-1-4799-6989-0 |s2cid=742206 }}</ref>
Compared to Funk MF, SVD++ takes also into account user and item bias.
The predicted rating user ''u'' will give to item ''i'' is computed as:
<math>\tilde{r}_{ui} = \mu + b_i + b_u + \sum_{f=0}^{n factors} H_{u,f}W_{f,i}</math>
SVD++ has however some disadvantages, with the main drawback being that this method is not ''model-based.'' This means that if a new user is added, the algorithm is incapable of modeling it unless the whole model is retrained. Even though the system might have gathered some interactions for that new user, its latent factors are not available and therefore no recommendations can be computed. This is an example of a [[Cold start (recommender systems)|cold-start]] problem, that is the recommender cannot deal efficiently with new users or items and specific strategies should be put in place to handle this disadvantage.<ref name="Kluver14">{{cite book|last1=Kluver |first1=Daniel |title=Proceedings of the 8th ACM Conference on Recommender systems - Rec ''Sys'' '14 |last2=Konstan |first2=Joseph A. |date=6 October 2014 |pages=121–128 |doi=10.1145/2645710.2645742 |publisher=ACM|chapter=Evaluating recommender behavior for new users |isbn=9781450326681 |s2cid=18509558 }}</ref>
A possible way to address this cold start problem is to modify SVD++ in order for it to become a ''model-based'' algorithm, therefore allowing to easily manage new items and new users.
As previously mentioned in SVD++ we don't have the latent factors of new users, therefore it is necessary to represent them in a different way. The user's latent factors represent the preference of that user for the corresponding item's latent factors, therefore user's latent factors can be estimated via the past user interactions. If the system is able to gather some interactions for the new user it is possible to estimate its latent factors.
Note that this does not entirely solve the [[Cold start (recommender systems)|cold-start]] problem, since the recommender still requires some reliable interactions for new users, but at least there is no need to recompute the whole model every time. It has been demonstrated that this formulation is almost equivalent to a SLIM model,<ref name="Zheng14">{{cite book |last1=Zheng |first1=Yong |last2=Mobasher |first2=Bamshad |last3=Burke |first3=Robin |title=CSLIM: contextual SLIM recommendation algorithms |date=6 October 2014 |pages=301–304 |doi=10.1145/2645710.2645756 |publisher=ACM|chapter=CSLIM |isbn=9781450326681 |s2cid=15931532 }}</ref> which is an [[item-item recommender|item-item]] model based recommender.
<math>\tilde{r}_{ui} = \mu + b_i + b_u + \sum_{f=0}^{n factors} \biggl( \sum_{j=0}^{n items} r_{uj} W_{j,f} \biggr) W_{f,i}</math>
With this formulation, the equivalent [[item-item recommender]] would be <math>\tilde{R} = R S = R W^{\rm T} W</math>. Therefore the similarity matrix is symmetric.
=== Asymmetric SVD ===
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