## 3 The Proposed Algorithm
Practically, this density increase effectively transforms the sparse CF datasets to dense, making it possible for other techniques (e.g., matrix factorization [38; 39], a technique showing relatively poor prediction accuracy when applied to sparse datasets [40; 41]) to be used to further improve prediction accuracy and/or recommendation quality. The proposed algorithm was also applied to a relatively dense dataset (the MovieLens 100 K, whose initial density was 3.17%) and has also been found to successfully increase both its density and coverage further, leading to additional improvement in the rating prediction accuracy. This indicates that the algorithm can be employed across all datasets, regardless of their densities. Hence, a CF system implementation may employ the proposed algorithm, without even examining the properties of the used dataset. Furthermore, the proposed algorithm achieved to increase CF rating prediction coverage by 36.5% on average when the PCC similarity metric was applied and by 24.1% on average when the CS similarity metric was applied. In regard to CF rating prediction accuracy, the CF\({}_{\text{DR}}\) algorithm achieved a MAE reduction of 2.06% and a RMSE reduction of 2.75%, when the PCC metric was applied. The respective average reductions for the CS similarity metric were 2.19% and 2.09%. Moreover, the CF\({}_{\text{DR}}\) algorithm, was also compared with (a) the CF\({}_{\text{VNN}}\) algorithm introduced in [27], where artificial user profiles (Virtual Near Neighbours--VNNs) are created by merging pairs of NN profiles corresponding to real users, and (b) the CF\({}_{\text{ROAF}}\) algorithm introduced in [29], where two users can be considered as NNs with each other if they are NNs to a third user. Both of these algorithms contribute to the \"grey sheep\" problem alleviation. The CF\({}_{\text{DR}}\) algorithm was found to consistently outperform both the CF\({}_{\text{VNN}}\) and the CF\({}_{\text{ROAF}}\) algorithms in all tested datasets and under both similarity metrics, for both rating prediction accuracy and coverage. A comparison was also made against state-of-the-art algorithms aiming to increase rating prediction coverage utilizing only the user-item rating matrix and has been shown to outperform all of them by a considerable performance margin. Regarding our future work, we plan to examine the proposed algorithm using additional correlation metrics, such as the Spearman coefficient, the Euclidian distance, and the Manhattan distance [59]. Adaptation of the proposed approach, so that it can be directly combined with MF techniques [38; 39], and exploit the benefits of merging estimators [11; 12; 13] will also be considered. Finally, the adaptation of the CF\({}_{\text{DR}}\) algorithm to take into account additional information, such as social network information, item (sub-)categories, and user characteristics and emotions [60; 61; 62; 63], will be also investigated. Conceptualization, D.M., D.S., G.K., and C.V.; Methodology, D.M., D.S., G.K., and C.V.; Software, D.M., D.S., and C.V.; Writing--original draft, D.M., D.S., G.K., and C.V.; Writing--review & editing, D.M., D.S., G.K., and C.V. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The authors declare no conflict of interest. ## References

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