# An Algorithm for Density Enrichment of Sparse Collaborative Filtering Datasets Using Robust Predictions as Derived Ratings

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## Abstract

**:**

_{DR}, is extensively evaluated using (1) seven widely-used collaborative filtering datasets, (2) the two most widely-used correlation metrics in collaborative filtering research, namely the Pearson correlation coefficient and the cosine similarity, and (3) the two most widely-used error metrics in collaborative filtering, namely the mean absolute error and the root mean square error. The evaluation results show that, by successfully increasing the density of the datasets, the capacity of collaborative filtering systems to formulate personalized and accurate recommendations is considerably improved.

## 1. Introduction

- Introducing the concepts of derived ratings (DRs) and robust predictions: derived ratings are created from predictions with a high level of confidence regarding their accuracy, which are termed as robust predictions. DRs complement the explicitly entered ratings in the user–item rating matrix, increasing the matrix density.
- Proposing an algorithm, termed CF
_{DR}, for the creation of DRs on the basis of real, explicit user ratings, while considering various prediction robustness criteria. - Extending the rating prediction computation algorithm to compute rating predictions on the complemented user-item rating matrix, while taking into account the level of confidence associated with each DR.

_{DR}algorithm, let us consider the case where we want to predict the rating r

_{U}

_{3,}

_{i}

_{4}that user U

_{3}would assign to item i

_{4}, in order to decide whether it should be recommended to him or not. Let us also assume that the RS computes user similarity using the Pearson correlation coefficient [1,2] and that, at rating prediction time, the user–item ratings database is as depicted in Table 1.

_{3}, i

_{4}), a prediction for r

_{U}

_{3,}

_{i}

_{4}cannot be computed, since: (1) U

_{3}has only one NN, U

_{2}(they share a positive correlation with each other, since they have both rated item i

_{2}with a high score), but U

_{2}has not rated item i

_{4}, and (2) although user U

_{1}has rated item i

_{4}, he does not belong to U

_{3}′s NN set (no common rated items). The case for cell (U

_{1}, i

_{3}) is similar while, notably, U

_{4}has not rated any item in common with other users, and i

_{5}has only been rated by U

_{4}, Hence, no rating prediction can be formulated for any item for U

_{4}, and no rating prediction can be computed for any user regarding item i

_{5}.

_{1}and U

_{2}are NNs with each other (they share a positive correlation, having both rated item i

_{1}with a low score) and hence if U

_{2}had asked for a prediction on item i

_{4}(instead of user U

_{3}) that prediction could have been formulated.

_{DR}algorithm proposed in this paper, a preprocessing step is executed that computes all the predictions that can be formulated for all the users in the CF rating database, and the predictions having a high level of confidence regarding their accuracy, i.e., the robust ones, are converted to DRs and stored into the user-item ratings database. Table 2 depicts the state of the user-item rating matrix from Table 1 after the execution of the preprocessing step of the CF

_{DR}algorithm (the newly inserted DRs are shown in bold).

_{U}

_{3,}

_{i}

_{4}and r

_{U}

_{3,}

_{i}

_{3}can now be formulated. The ability to compute rating predictions that could not be formulated on the basis of the original user-item rating matrix stems from the two following properties of the DR-enriched -item rating matrix:

_{U}

_{3,}

_{i}

_{2}and r

_{U}

_{3,}

_{i}

_{1}in the DR-enriched database depicted in Table 2, U

_{3}′s near neighbourhood has extended to include U

_{1}, hence any rating recorded by U

_{1}can now be used for computing rating predictions for U

_{3}, as is the case with r

_{U}

_{3,}

_{i}

_{4}. It is worth noting at this point that the ability to compute a prediction for r

_{U}

_{3,}

_{i}

_{4}can lead to the formulation of a personalized recommendation for user U

_{3}: in the situation depicted in Table 1, only the value of r

_{U}

_{3,}

_{i}

_{1}could be predicted. However, this value is very low (1) and hence i

_{1}would not be recommended to U

_{3}. Then, the recommendation algorithm would either refrain from formulating a recommendation or use the mean database rating value for items i

_{4}and i

_{5}, which is 5 for both items, and therefore would arbitrarily choose among them. In the DR-enriched rating matrix context, the recommendation algorithm has a prediction for rating r

_{U}

_{3,}

_{i}

_{4}at its disposal and can therefore knowledgably select and recommend i

_{4}to U

_{3}.

_{U}

_{3,}

_{i}

_{3}, user U

_{2}, who was formerly a NN of U

_{1}, can now contribute to the computation of a rating prediction for r

_{U}

_{3,}

_{i}

_{3}. Regarding the recommendation formulation stage, when the RS utilises the DR-enriched rating matrix the recommendation algorithm, having at its disposal a prediction for rating r

_{U}

_{3,}

_{i}

_{3}, it can knowledgeably refrain from recommending item i

_{3}to U

_{1}, rather than relying on the database rating mean value for this item.

_{4}, who had no common ratings with any other user in the user-item rating matrix, still no near-neighbours can be identified for him/her in the context of the DR-enriched user-item rating matrix. Similarly, in the same context, no personalized prediction can be computed for item i

_{5}. These cases mainly correspond to the cold start problem [4,5]. The cold start problem can be tackled using content-based algorithms [6,7], context-based algorithms [8], and social network-based algorithms [9,10]. Each class of such algorithms requires additional information, while it is also possible that such algorithms are combined with the approach presented in this paper.

_{DR}algorithm, an extensive evaluation is conducted, assessing (1) the degree of density enrichment, (2) the prediction coverage increase, and (3) the prediction accuracy increase, achieved by the proposed algorithm. In this evaluation, we use seven contemporary and widely used CF datasets from various domains (videogames, music, movies, food, etc.). We consider the two most widely used CF similarity metrics, namely the Pearson correlation coefficient and the cosine similarity and the two most widely used CF error metrics, namely the mean average error and the root mean square error.

_{DR}algorithm, proposed in this paper, can be combined with other works in the domain of CF, targeting either to (1) enhance rating prediction accuracy [14,15,16,17], (2) improve recommendation quality [18,19,20,21], (3) increase rating prediction computation efficiency [22,23,24,25], or (4) further increase prediction coverage in CF-based RSs [26,27,28,29].

_{DR}algorithm. Section 4 reports on the methodology for tuning the algorithm operation and evaluates the presented algorithm. Finally, Section 5 discusses the experimental results and presents a comparison with state-of-the-art algorithms addressing the sparsity issue, while Section 6 concludes the paper and outlines future work.

## 2. Related Work

_{negNNs}algorithm presented in [26] increases the CF rating prediction coverage in sparse datasets, by incorporating, in the rating prediction computation process, users with negative similarity to the user for whom the rating prediction is computed. Margaris et al. [27] propose the CF

_{VNN}algorithm, which creates virtual user profiles, termed virtual NNs (VNNs), by merging pairs of NN profiles corresponding to real users having high similarity. The introduction of the VNN profiles contributes to the alleviation of the “grey sheep” problem. Additionally, Margaris and Vassilakis [29] present an algorithm, termed CF

_{FOAF}, which adopts the friend-of-a-friend concept in CF, in the sense that two users can be considered as NNs with each other if they are NNs to a third user. Effectively, the CF

_{FOAF}algorithm explores the effect of considering transitivity of the NN relationships, examining different parameters of computing properties of the derived relationships.

_{DR}, has been validated using seven contemporary and widely used CF datasets and has been found to significantly upgrade the density of the sparse CF datasets and the CF rating prediction coverage. At the same time, CF

_{DR}successfully increases the CF rating prediction accuracy. This behaviour is proven to be consistent under both similarity metrics tested.

## 3. The Proposed Algorithm

- Firstly, for each user U, his NNs, i.e., the set of users that have similar tastes to U and who will act as recommenders to U, are identified. Typically, in order for a user V to belong to U’s NN set, the similarity of ratings between U and V must exceed a specified threshold (e.g., the value 0, when using the PCC metric). The NN set for a user U may also be limited to the top-k users that have the higher similarity metrics to user U.
- Subsequently, predictions for the ratings that user U would enter for items that the user has not already rated are computed, based on the ratings of the user’s NNs for the same items.

_{U,i}, which user U would assign to item i, is computed as shown in Equation (3):

_{DR}algorithm, proposed in this paper, introduces a preprocessing step where, for each user U, rating predictions for each of the items that s/he has not already rated are computed using Equation (3). Obviously, this is only possible for items that have been rated by at least one user within a U’s NN set. Such prediction is examined for robustness, i.e., for high confidence about its accuracy. If the prediction is deemed robust, then the (robust) prediction rp

_{U,i}is stored in the user-item rating database as a DR, dr

_{U,i}. The DRs stored in the rating database can then be used in the main operation cycle of the user-user CF algorithm, thus contributing to (a) the phase of computing each user’s NN set and (b) the phase of calculating the rating predictions. The incorporation of DRs in the user-item rating database increases the density of the database, hence extending the users’ NN sets and increasing coverage, i.e., widening the range of user/item combinations for which personalized predictions can be computed.

_{U,i}are:

_{U,i}exceeds a threshold thr

_{NN}: The rationale behind the introduction of this criterion is that, if a rating prediction has been formulated by synthesizing the opinion of a large number of users, then this prediction is deemed more probable to be accurate. Formally, this criterion is expressed as $\left|\left\{\mathrm{V}\in \mathrm{NN}\left(\mathrm{U}\right):{\mathrm{r}}_{\mathrm{V},\mathrm{i}}\in R\right\}\right|\ge th{r}_{NN}$, where R is the extension of the user-item rating database. The optimal value for $th{r}_{NN}$ is explored experimentally in Section 4.

_{U,i}exceeds a threshold thr

_{sim}: The rationale behind the introduction of this criterion is that, if a rating prediction has been formulated by synthesizing the opinion of users that are “more similar” to U, then this prediction is deemed more probable to be accurate. Formally, this criterion is expressed as $\sum}_{\mathrm{V}\in \mathrm{NN}\left(\mathrm{U}\right):{\mathrm{r}}_{\mathrm{V},\mathrm{i}}\in R}sim\left(U,V\right)\ge th{r}_{sim$. Again, the optimal value for $th{r}_{sim}$ is explored experimentally in Section 4.

_{U,i}based on the DR-enriched user-item rating database, both explicitly entered ratings and DRs are considered for the calculation of the value of p

_{U,i}, as shown in Equation (3). Taking into consideration, though, that DRs are estimates on the user ratings, as contrasted to explicitly entered ratings which constitute factual data, it may be appropriate to assign to DRs a lower weight when compared to explicitly entered ratings for this calculation. To this end, the proposed algorithm CF

_{DR}modifies Equation (3) as follows:

_{V,i}) denotes the weight assigned to each NN user’s rating taking part in the prediction formulation, depending on the type of this rating (real/explicit vs. derived, i.e., produced during the CF

_{DR}algorithm’s preprocessing step). Formally, the weight w(r

_{V,i}) of a rating r

_{V,i}is computed as shown in Equation (5):

_{dr}≤ 1. The optimal value for parameter w

_{dr}is experimentally determined in Section 4.

_{DR}algorithm that creates the DRs and the DR-enriched version of the user-item rating database according to Equations (3) and (5), while Algorithm 2 presents the prediction computation step of the CF

_{DR}algorithm, according to Equation (4).

_{DR}algorithm, the parameters thr

_{NN}, thr

_{sim}and w

_{dr}, listed above, need to be determined. The optimal values of these parameters are explored experimentally in the following section, along with the evaluation of the CF

_{DR}algorithm.

Algorithm 1. Implicit ratings creation and storage, produced by the CFDR algorithm’s preprocessing step. |

PROCEDURE enrich_Rating_DB (Rating_DB R, int thr_{NN}, float thr_{sim})/* Implementation of the preprocessing step of the CF _{DR}, during which DRs are computed.INPUT: R is the initial User-Item Rating Database; thr _{NN} and thr_{sim} are the thresholds for theminimum number of NNs contributing to the formulation of a prediction and their cumulative similarity, respectively, used in the prediction robustness evaluation criteria. OUTPUT: An enriched version of the User-Item Rating Database, including the derived ratings */ DR_set = ∅ /* DR_set is the set of dynamic ratings to be added to the ratings matrix */ FOREACH user U FOREACH item i IF (r _{U,i} == NULL) THEN/* User U has not rated item i; predict the rating according to Equation (3) */ rating_prediction = $\overline{{r}_{U}}+\frac{{{\displaystyle \sum}}_{V\in N{N}_{U}}sim\left(U,\text{}V\right)\times \left({r}_{V,i}-\overline{{r}_{V}}\right)}{{{\displaystyle \sum}}_{V\in N{N}_{U}}sim\left(U,\text{}V\right)}$ /* Test the rating prediction for robustness, by applying the threshold criteria */ IF ((NN_counter(rating_prediction) ≥ thr _{NN}) ∧ (NN_sim(rating_prediction) ≥ thr_{sim}))/* Compute the weight of the dynamic rating according to Equation (5) and add the rating in DR_set */ value(dr _{U,i}) = rating_predictionweight(dr _{U,i}) = w(dr_{U,i})DR_set = DR_set ∪ {dr _{U,i}}ENDIF ENDIF END /* FOREACH item i */ END /* FOREACH user U */ /* Finally append all DRs into the ratings database */ R = R ∪ DR_set END /* PROCEDURE */ |

Algorithm 2. Prediction computation under the CFDR algorithm. |

FUNCTION Compute_Prediction (enhanced_Rating_DB R, User U, Item i) /* Implementation of formula (4), used for the prediction computation INPUT: U is the user for whom the prediction will be computed; i is the respective item OUTPUT: the rating prediction value, or NULL if no NN of U has rated it (hence a prediction cannot be computed) */ IF (∃ V∈ NN _{U}: r_{V,i} ≠ NULL) THEN/* compute the prediction according to equation (4) */ prediction = $\overline{{r}_{U}}+\frac{{{\displaystyle \sum}}_{V\in N{N}_{U}}sim\left(U,\text{}\mathrm{V}\right)\times w\left({r}_{V,i}\text{}\right)\times \left({r}_{V,i}-\overline{{r}_{V}}\right)}{{{\displaystyle \sum}}_{V\in N{N}_{U}}sim\left(U,\text{}V\right)*w\left({r}_{V,i}\right)}$ ELSE prediction = NULL ENDIF RETURN prediction END /* FUNCTION */ |

## 4. Algorithm Tuning and Experimental Evaluation

- identify the optimal settings for the parameters thr
_{NN}, thr_{sim}and w_{dr}, in order to tune the CF_{DR}algorithm, and - assess the CF
_{DR}algorithm performance in terms of density enrichment, rating prediction coverage and accuracy increase.

- they are widely used as benchmarking datasets in CF research;
- they belong to multiple item domains (videogames, office supplies, food, music, movies, etc.);
- they are contemporary (published between 1996 and 2018);
- they differ in size (ranging from 1.4 MB to 40 MB in plain text format), user count (ranging from 670 to 124 K), item count (ranging from 2.4 K to 64 K) and density (ranging from 0.023% to 3.17%).

_{DR}algorithm. In this experiment, accuracy was measured by comparing the actual ratings in the validation portion of the dataset against the corresponding ratings that were computed by the two algorithms (DR-enriched under different hyperparameter settings, as well as baseline). The prediction coverage was computed as the number of unknown ratings for which rating predictions could be computed to the total number of unknown ratings.

_{DR}algorithm, while accuracy was measured against the ratings in the holdout portion. The parameters for the plain CF algorithm were set as follows: user V was considered an NN of U if sim(U, V) > 0 and a prediction p

_{U,i}was formulated when at least one NN of U had entered a rating for item i.

#### 4.1. Determining the Algorithm’s Parameters

_{NN}, thr

_{sim}, and w

_{dr}, used in the CF

_{DR}algorithm. In order to find the optimal setting for these parameters, we explored different combinations of values for them. In total, more than 30 candidate setting combinations were examined. For conciseness, our presentation is confined to the most indicative settings. For each of the parameter combinations, we present the coverage increase and the improvement of the rating prediction accuracy that were achieved (expressed as reduction to the MAE and RMSE metrics). The results were found to be relatively consistent across the seven datasets tested (for the majority of tests, the parameter value combinations were ranked in the same order across all datasets). Consequently, in this subsection, only the average values of the respective coverage increase and error metrics for the seven datasets are presented.

_{NN}, thr

_{sim}and w

_{dr}, when similarity is measured using the PCC metric and using the performance of the plain CF algorithm as a baseline. Similarly, Figure 2 depicts the respective measures under the same parameter value combinations, when similarity is measured using the CS metric, again using the performance of the plain CF algorithm as a baseline.

_{NN}= 3, thr

_{sim}= 0.0, w

_{dr}= 1.0}) is the best performer across both error metrics, with its harmonic mean being equal to 0.84 under both error metrics. The fourth setting ({thr

_{NN}= 3, thr

_{sim}= 2.5, w

_{dr}= 0.8}) is the runner up, scoring an HM equal to 0.82 under the MAE error metric and 0.80 under the RMSE error metric.

_{NN}= 3, thr

_{sim}= 0.0, w

_{dr}= 1.0}) attains the highest harmonic means under both error metrics (0.83 under both the MAE and RMSE), closely followed by the fourth setting (0.82 under the MAE and 0.81 under the RMSE). The fifth setting ({thr

_{NN}= 2, thr

_{sim}= 1.0, w

_{dr}= 1.0}) follows, with its HM under the MAE being 0.81 and its HM under the RMSE being 0.80.

_{NN}= 3, thr

_{sim}= 0.0, w

_{dr}= 1.0}) achieves the best performance under both similarity metrics and under both error metrics, for the rest of this paper we adopt this setting for all experiments. This practically means that (a) any prediction with at least 3 NNs taking part in its formulation is considered robust and hence produces a DR and (b) the importance of DRs in the prediction formulation process is equal to that of an explicitly entered (real) user rating. This aspect is also exploited for optimization purposes, since no additional information needs to be entered to the user-rating matrix in order to discriminate between real ratings and DRs.

_{DR}algorithm parameters were set to the values proven to be optimal (i.e., {thr

_{NN}= 3, thr

_{sim}= 0.0, w

_{dr}= 1.0}).

#### 4.2. Performance Evaluation

_{DR}algorithm, we evaluated the performance of the algorithm in terms of dataset density increase, prediction coverage increase, and prediction accuracy improvement using the seven datasets listed in Table 4.

#### 4.2.1. Density Increase

_{DR}algorithm’s preprocessing step. Note that a base-10 logarithmic scale is used for the Y axis.

_{DR}algorithm preprocessing step, all the datasets, even the more sparse ones (such as the Amazon “CDs and Vinyl”, having an initial density of 0.023%) exceed the density threshold of 1% set by [56], beyond which a dataset D is considered to be dense.

#### 4.2.2. Prediction Coverage Increase

_{DR}algorithm presented in this paper, we perform a detailed comparison between its performance and the performance of the following algorithms:

- The CF
_{VNN}algorithm [27], where artificial user profiles (virtual near neighbours—VNNs) are created by combining pairs of real users which are NNs of high similarity. The CF_{VNN}algorithm was configured to use the optimal parameters for its operation reported in [27], i.e., Th(sim) = 1.0 and Th(cr) = 1. For more details on the parameters, the interested reader is referred to [27]. - The CF
_{FOAF}algorithm [29], where the transitivity of the NN relationship is explored, and two users can be considered as NNs with each other if they are NNs to a third user. The CF_{FOAF}algorithm was configured to use the optimal parameters for its operation reported in [29], i.e., Th(sim, src) = 0, Th(sim, targ) = 0, Th(sim, endpoints) = 0, Th(cr, src) = 1, Th(cr, targ) = 1, and Th(cr, comb) = 2. For more details on the parameters, the interested reader is referred to [29].

_{VNN}algorithm [27] was presented in 2019, while the CF

_{FOAF}algorithm was presented in 2020) algorithms targeting primarily to increase the CF rating prediction coverage and also to alleviate the “grey sheep” problem, (b) achieve considerable improvement in coverage while additionally attaining slight improvement on rating prediction accuracy, and (c) necessitate no supplementary data, such as user relationships obtained from social networks, item categories, etc.

_{VNN}and CF

_{FOAF}algorithms, for the performance comparison, is that among all CF algorithms targeting coverage increase without necessitating additional data, the first algorithm is proven to be better at rating prediction accuracy, while the second one achieves higher rating prediction coverage increments. Therefore, by comparing the proposed algorithm with both the CF

_{VNN}and CF

_{FOAF}algorithms, we obtain a holistic view of the algorithm performance, covering both rating prediction aspects (coverage and accuracy).

_{DR}algorithm, presented in this paper, successfully increases the average coverage across all datasets by 36.52% against the baseline (the coverage increment is calculated as $\frac{coverage\left(DR-enriched\text{}dataset\right)-coverage\left(baseline\right)}{coverage\left(baseline\right)}$). On the individual dataset level, the coverage improvement ranges from 2.61%, for the MovieLens 100 K dataset, to 83.1%, for the Amazon “Grocery and Gourmet Food” dataset. Notably, the MovieLens 100 K is a dense dataset (its density is equal to 3.17%), consequently exhibiting very high coverage in its initial form (approximately equal to 94%), hence the coverage improvement margin is limited.

_{FOAF}algorithm, introduced in [29], achieved an average coverage increase of 34.7%, while the CF

_{VNN}algorithm [27] achieved an average coverage increase of 30.1%.

_{FOAF}algorithm, which was found to achieve higher prediction coverage than the CF

_{VNN}algorithm, ranges from 4.1%, for the Amazon “Grocery and Gourmet Food” dataset, to 8.1%, for the Amazon “Movies & TV” dataset, for the sparse (Amazon) datasets that were considered. The average performance edge, across all the sparse datasets, of the proposed algorithm against the CF

_{FOAF}algorithm was measured at 5.44%, while the respective edge against the CF

_{VNN}algorithm was measured at 25.9%.

_{VNN}algorithm by a margin of 0.47% in absolute figures, while, in terms of relative improvement, the edge of the proposed algorithm is quantified to 22%.

_{DR}algorithm, presented in this paper, achieves an increase of the average coverage across all datasets by 24.13% against the baseline. On the individual dataset level, the coverage improvement ranges from 1.59%, for the MovieLens 100 K dataset, to 61.12%, for the Amazon “Grocery and Gourmet Food” dataset. Again, the smallest coverage increase is achieved in the case of the MovieLens 100 K, which, due to its high density and the corresponding high initial coverage, offers little coverage improvement potential.

_{FOAF}algorithm, which was again found to achieve higher prediction coverage than the CF

_{VNN}algorithm, ranges from 1.3%, for the Amazon “Grocery and Gourmet Food” dataset, to 4.4%, for the Amazon “Movies and TV” dataset. The average performance edge, across all the sparse datasets, of the proposed algorithm against the CF

_{FOAF}algorithm was measured at 2.6%, while the respective edge against the CF

_{VNN}algorithm was measured at 14.8%.

_{FOAF}algorithm by a margin of 0.21% in absolute figures, while, in terms of relative improvement, the edge of the proposed algorithm is quantified to 15.2%.

#### 4.2.3. Prediction Accuracy Increase

_{DR}algorithm, we also compare its performance to the performance of the CF

_{VNN}algorithm introduced in [27] and the CF

_{FOAF}algorithm introduced in [29].

_{DR}algorithm, presented in this paper, achieves an average prediction MAE reduction equal to 2.06% (in absolute terms 0.877 versus 0.895, the plain CF achieves), exceeding the performance of the CF

_{VNN}algorithm [27] (which is measured at 1.42%) by 0.64% in absolute figures, or 45.1% as a relative improvement. When compared with the CF

_{FOAF}algorithm [29], the CF

_{DR}algorithm outperforms CF

_{FOAF}(whose MAE reduction was measured at 1.17%) by a margin of 76.3%.

_{VNN}algorithm (which achieves a higher MAE reduction than the CF

_{FOAF}algorithm), ranges from 25.9% (for the Amazon “Digital Music” dataset) to 84.4% (for the Amazon “Videogames” dataset).

_{DR}algorithm, presented in this paper, achieves an average prediction MAE reduction equal to 2.19% (in absolute terms 0.814 versus 0.832, the plain CF achieves), exceeding the performance of the CF

_{VNN}algorithm [27] (which is measured at 1.65%) by 0.54% in absolute figures, or 32.1% as a relative improvement. The CF

_{DR}algorithm also surpasses the performance of the CF

_{FOAF}algorithm [29] (which is measured at 1.55%) by 41.4%.

_{VNN}algorithm (which achieves higher MAE reduction than the CF

_{FOAF}algorithm), ranges from 8.12% (for the Amazon “Digital Music” dataset) to 79.7% (for the Amazon “Office Supplies” dataset).

_{DR}algorithm, presented in this paper, achieves an average prediction RMSE reduction equal to 2.75% (in absolute terms 1.229 versus 1.264, the plain CF achieves), exceeding the performance of the CF

_{VNN}algorithm [27] (which is measured at 2.09%) by 0.66% in absolute figures, or 31.5% as a relative improvement. In comparison to the CF

_{FOAF}algorithm [29], the CF

_{DR}algorithm surpasses CF

_{FOAF}(measured at 1.46%) by 88%. Similarly to the case of the MAE, listed above, the considerable differences observed for the values of the MAE metric between the MovieLens dataset on the one hand and the Amazon datasets on the other is that in the MovieLens dataset the ratings range from 0 to 9, whereas in the Amazon datasets ratings range from 1 to 5.

_{VNN}algorithm, which achieves a higher RMSE reduction than the CF

_{FOAF}algorithm, ranges from 17.5% (for the Amazon “CDs and Vinyl” dataset) to 88.6% (for the Amazon “Office Supplies” dataset).

_{DR}algorithm, presented in this paper, achieves an average prediction RMSE reduction equal to 2.09% (in absolute terms 1.151 versus 1.176, the plain CF achieves), exceeding the performance of the CF

_{VNN}algorithm [27] (which is measured at 1.48%) by 0.61% in absolute figures or by 41.5% as a relative improvement. The CF

_{DR}algorithm also outperforms the CF

_{FOAF}algorithm [29], whose performance is measured at 1.47%, by 42.2%. On the individual dataset level, the performance edge of the proposed algorithm against the CF

_{VNN}algorithm ranges from 8.95% (for the Amazon “Digital Music” dataset) to 87.5% (for the Amazon “Grocery and Gourmet Food” dataset).

## 5. Discussion of the Results and Comparison with Other Works

_{DR}algorithm, presented in this paper, can successfully increase the density of the CF datasets, regardless of their initial density, since it was successfully applied to both sparse and dense datasets. At the same time, the CF

_{DR}algorithm can achieve both a prediction coverage improvement and a significant prediction error reduction, for both the error metrics tested. In regard to the DR formulation, which is performed within the CF

_{DR}preprocessing step, the optimal setting for considering a prediction as robust (and hence create a DR) is when at least three NNs of a user take part in the prediction formulation. Furthermore, the optimal significance of the DRs was found to be exactly the same with the real (explicit) user ratings.

_{DR}algorithms against other algorithms published in the literature and target the increase of coverage of CF-based algorithms. To provide a fair comparison basis, the algorithms included in the comparison are limited to those which utilize only the user-item rating matrix, without necessitating the availability of additional data, such as social relationships [24,36] or trust [57,58].

_{DR}algorithm achieves higher coverage increase than all other state-of-the-art algorithms, both for the cases of sparse and dense datasets. This is also true for the rating prediction accuracy improvement. More specifically, in sparse datasets, the CF

_{DR}algorithm outperforms other algorithms in the prediction coverage by a margin ranging from 1.7% to 6.6% in absolute numbers (or 4.9% to 21.7% as a relative increase), observed for the CF

_{FOAF}[29] and CF

_{VNN}[27] algorithms, respectively. The corresponding accuracy improvement, as quantified by the MAE metric, varies from 0.71% to 0.98% in absolute numbers (or 45.5% to 80% as a relative increase), observed for the CF

_{VNN}[27] and CF

_{FOAF}[29] algorithms, respectively. It is worth noting that the papers cited in Table 5 employ different techniques to quantify coverage increase and rating prediction accuracy: [33] and [45] use a five-fold cross-validation, while [27] and [29] employ the hide-one technique. For comparison fairness purposes, we ran additional experiments to compute the coverage increase and the error metrics using the five-fold cross-validation and the hide-one techniques, and when comparing the CF

_{DR}with another algorithm, we used the results obtained using the same technique employed in the original algorithm publication. In all additional experiments, the CF

_{DR}hyperparameters were set to the best performing composition, i.e., {thr

_{NN}= 3, thr

_{sim}= 0.0, w

_{dr}= 1.0}.

_{DR}algorithm outperforms other algorithms in coverage increase by a margin ranging from 0.59% to 0.64% in absolute numbers (or 28.5% to 33.7% as a relative increase), as observed for the HyCov [33] and CF

_{FOAF}[29] algorithms, respectively. The corresponding accuracy improvement, as quantified by the MAE metric, varies from 0.15% to 0.54% in absolute numbers (or 32.6% to 450% as a relative increase), observed for the CF

_{VNN}[27] and the HyCov [33] algorithms, respectively.

## 6. Conclusions and Future Work

_{DR}algorithm, which is a novel CF algorithm for enhancing the density of sparse CF datasets. The presented algorithm introduced the concept of derived ratings, which are ratings formulated by robust predictions. The experimental results indicated that the optimal setting for considering a prediction as robust (and hence create a derived rating) is when at least three of a user’s NNs take part in the prediction formulation. The experimental results indicated that the optimal significance of the derived ratings, created by the CF

_{DR}algorithm preprocessing step, was the exact same with the real user ratings. Therefore, no other complementary information needs to be stored in the user-item rating DB, except the DRs.

_{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%.

_{DR}algorithm, was also compared with (a) the CF

_{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

_{FOAF}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

_{DR}algorithm was found to consistently outperform both the CF

_{VNN}and the CF

_{FOAF}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.

_{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.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Prediction error reduction and coverage increase under different parameter value combinations, when using the PCC as the similarity metric.

**Figure 2.**Prediction error reduction and coverage increase under different parameter value combinations, when using the CS as the similarity metric.

**Figure 3.**Harmonic Mean of the normalized accuracy and coverage increase measures for the candidate setting, when using the PCC as the similarity metric.

**Figure 4.**Harmonic Mean of the normalized accuracy and coverage increase measures for the candidate setting, when using the CS as the similarity metric.

**Figure 6.**Coverage increase for the datasets summarized in Table 4, when the PCC user similarity metric is used.

**Figure 7.**Coverage increase for the datasets summarized in Table 4, when the CS user similarity metric is used.

**Figure 8.**Rating prediction MAE for the datasets summarized in Table 4, when the PCC user similarity metric is used.

**Figure 9.**Rating prediction MAE for the datasets summarized in Table 4, when the CS user similarity metric is used.

**Figure 10.**Rating prediction RMSE for the datasets summarized in Table 4, when the PCC user similarity metric is used.

**Figure 11.**Rating prediction RMSE for the datasets summarized in Table 4, when the CS user similarity metric is used.

User/Item | i_{1} | i_{2} | i_{3} | i_{4} | i_{5} |
---|---|---|---|---|---|

U_{1} | 1 | 5 | |||

U_{2} | 1 | 5 | |||

U_{3} | 5 | 1 | |||

U_{4} | 5 |

**Table 2.**User-item rating database excerpt after the execution of the preprocessing step of CF

_{DR.}

User/Item | i_{1} | i_{2} | i_{3} | i_{4} | i_{5} |
---|---|---|---|---|---|

U_{1} | 1 | 5 | 5 | ||

U_{2} | 1 | 5 | 1 | 5 | |

U_{3} | 1 | 5 | 1 | ||

U_{4} | 5 |

Notation | Description |
---|---|

r_{U,i} | The rating explicitly entered by user U for item i |

p_{U,i} | The prediction computed for the rating that user U would enter for item i |

$\overline{{\mathrm{r}}_{\mathrm{U}}}$ | The mean value of ratings entered by user U |

NN_{U} | The set of near neighbours for user U |

sim(U,V) | The similarity between users U and V under the employed similarity metric |

dr | Derived rating |

thr_{NN} | The minimum number of NNs that a rating prediction should be based on, in order to be considered “robust” and added as a derived rating |

thr_{sim} | The minimum cumulative similarity between a user U and his/her NNs that contributed to the calculation of some rating prediction, in order for the rating to be considered “robust” and added as a derived rating |

w_{dr} | The weight assigned to derived ratings |

Dataset Name | #Users | #Items | #Ratings | Avg. #Ratings/User | Density | DB Size (in Text Format) |
---|---|---|---|---|---|---|

Amazon “Videogames” [51] | 24 K | 11 K | 232 K | 9.7 | 0.089% | 5 MB |

Amazon “CDs and Vinyl” [51] | 75 K | 64 K | 1.1 M | 14.7 | 0.023% | 25 MB |

Amazon “Movies and TV” [51] | 124 K | 50 K | 1.7 M | 13.7 | 0.027% | 40 MB |

Amazon “Digital Music” [51] | 5.5 K | 3.5 K | 65 K | 11.8 | 0.327% | 1.4 MB |

Amazon “Office Supplies” [51] | 4.9 K | 2.4 K | 53 K | 10.8 | 0.448% | 1.1 MB |

Amazon “Grocery and Gourmet Food” [51] | 15 K | 8.7 K | 151 K | 10.1 | 0.118% | 3.4 MB |

MovieLens “Latest 100 K—Recommended for education and development” [53] | 670 | 4.7 K | 100 K | 149.3 | 3.17% | 2.2 MB |

**Table 5.**Comparison of CF

_{DR}with algorithms that exploit only the user-item rating matrix for achieving coverage increase.

Algorithm | Evaluation Dataset(s) | Coverage Increase | Accuracy Increase | Comments |
---|---|---|---|---|

HyCov [33] | MovieLens | 2.07% | MAE: 0.12% | The CF_{DR} algorithm achieves higher coverage and accuracy improvements as measured using the MAE metric in the same dataset (2.66% and 0.66%, respectively, under a five-fold cross validation performance assessment procedure). |

CF_{VNN} [27] | MovieLens & Amazon | avg. for sparse datasets: 30.54% MovieLens: 1.90% | avg. for sparse datasets: MAE: 1.56%; RMSE: 2.33% MovieLens: MAE: 0.46%; RMSE: 0.56% | The CF_{DR} algorithm achieves higher coverage and accuracy improvements both for sparse datasets (avg. coverage increase: 37.17%, avg. MAE improvement: 2.27%; avg. RMSE improvement: 3.04%, under a hide-one performance assessment procedure) and the dense MovieLens dataset (2.54%, 0.61% and 0.67%, respectively, again under a hide-one performance assessment procedure). |

CF_{FOAF} [29] | MovieLens & Amazon | avg. for sparse datasets: 35.45% MovieLens: 1.90% | avg. for sparse datasets: MAE: 1.29%; RMSE: 1.67% MovieLens: MAE: 0.36%; RMSE: 0.25% | The CF_{DR} algorithm achieves higher coverage and accuracy improvements both for sparse datasets (avg. coverage increase: 37.17%, avg. MAE improvement: 2.27%; avg. RMSE improvement: 3.04%, under a hide-one performance assessment procedure) and the dense MovieLens dataset (2.54%, 0.61% and 0.67%, respectively, again under a hide-one performance assessment procedure). |

ANLF [45] | MovieLens & Jester | None reported; the algorithm is based on matrix factorization, hence a value is predicted for all items for all users | avg. for the MovieLens & Jester datasets: RMSE 0.25% and 0.47%, respectively | CF_{DR} algorithm achieves higher accuracy improvements than ANLF (MovieLens dataset RMSE improvement: 0.71% under a five-fold cross validation performance assessment procedure). Note that, since the ANLF is matrix factorization based, a prediction is always formulated for every empty cell in the user-item rating matrix, with predictions involving users or items having very few ratings degenerating to a dataset-dependent constant value [40]. Despite this particularity, accuracy precisions are comparable because they are always computed considering solely known ratings, because only for these ratings the ground truth is known. |

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**MDPI and ACS Style**

Margaris, D.; Spiliotopoulos, D.; Karagiorgos, G.; Vassilakis, C. An Algorithm for Density Enrichment of Sparse Collaborative Filtering Datasets Using Robust Predictions as Derived Ratings. *Algorithms* **2020**, *13*, 174.
https://doi.org/10.3390/a13070174

**AMA Style**

Margaris D, Spiliotopoulos D, Karagiorgos G, Vassilakis C. An Algorithm for Density Enrichment of Sparse Collaborative Filtering Datasets Using Robust Predictions as Derived Ratings. *Algorithms*. 2020; 13(7):174.
https://doi.org/10.3390/a13070174

**Chicago/Turabian Style**

Margaris, Dionisis, Dimitris Spiliotopoulos, Gregory Karagiorgos, and Costas Vassilakis. 2020. "An Algorithm for Density Enrichment of Sparse Collaborative Filtering Datasets Using Robust Predictions as Derived Ratings" *Algorithms* 13, no. 7: 174.
https://doi.org/10.3390/a13070174