ER-SCoR: An Equal Ratings Impact-Based Recommender System Using Synthetic Coordinates
CostasPanagiotakis1✉,3Emailcpanag@hmu.gr
HarrisPapadakis2✉Emailadanar@hmu.gr
ParaskeviFragopoulou2,3Emailfragopou@ics.forth.gr
1
A
Department of Management Science and TechnologyHellenic Mediterranean UniversityLakonia72100Agios NikolaosCreteGreece2Department of Electrical and Computer EngineeringHellenic Mediterranean UniversityEstavromenos71410HeraklionCreteGreece
3Institute of Computer ScienceFoundation for Research and Technology-HellasVassilika Vouton70013HeraklionCreteGreece
A
Abstract
In this article, we introduce ER-SCoR, an Equal Ratings impact-based Recommender System built upon Synthetic Coordinates, which is shown to outperform the state-of-the-art algorithmic techniques as well as the original Synthetic Coordinate based Recommendation system (SCoR). SCoR assigns a set of synthetic coordinates to every node (both users and items), such as the distance between a user and an item corresponds to an accurate prediction of the user’s preference for that item. ER-SCoR enhances this model by (i) enforcing equal contributions from all ratings during coordinate updates, and (ii) incorporating three additional terms into the recommendation process: a global system belief, a user-specific belief, and an item-specific belief. These modifications constitute fundamental changes in the core system architecture and improve convergence speed, accuracy, and stability. ER-SCoR preserves the advantages of SCoR like parameter-free configuration, robustness to cold-start problems, and linear computational complexity, while achieving faster convergence and improved predictive performance.Extensive experiments across five real-world datasets demonstrate that ER-SCoR consistently yields lower RMSE compared to existing approaches, and provides meaningful dataset annotations, including identification of outliers, users with similar preferences and items that receive similar user ratings.
Keywords
Recommender System
Matrix Factorization
Synthetic Coordinates
Personalized Recommendations
MSC Classification
68T20
68W50
A
Introduction
Recommender Systems (RS) collect user preference data—either through explicit ratings or by monitoring user behavior across various sources—to generate personalized predictions and item suggestions bod13, score, he2017neural, dtec. Increasing research interest in the field has led to a similarly wide range of proposed approaches, leveraging a diverse spectrum of techniques \cite{xie2016user}. The core task of a RS is to predict missing ratings for user-item pairs. Formally, given a set of users, a set of items (e.g., movies, products, songs), and a set of known user-item ratings
, the system aims to estimate ratings for pairs not included in
. In a recommender system, users typically create accounts and rate items, allowing the system to learn their preferences. As the number of users and ratings grows, prediction accuracy generally improves. However, in cases of data sparsity or lack of information about new users—known as the sparsity and cold start problems, respectively—prediction quality often degrades. The core function of RS is to predict a user’s preference for an item. Even for a-priori known ratings on items, model-based approaches often exhibit prediction errors, as they cannot fully capture the underlying rating patterns. These inaccuracies stem from the system’s degrees of freedom and the dimensionality reduction required to generalize learned knowledge into predictions \cite{dtec}.
In our previous work, a Synthetic Coordinates based Recommender System (SCoR) has been described \cite{score}. This approach is based on assigning synthetic coordinates to users and items, so that the the proximity between a user and an item accurately forecasts the user's preference for that item. SCoR updates the synthetic coordinates of nodes by randomly traversing the list of nodes.
In this work, we propose a recommender system (ER-SCoR) based on a Synthetic Coordinates based recommender system under the Equal Ratings Impact principle that improves the results of SCoR with faster convergence. ER-SCoR updates the synthetic coordinates of nodes in a more efficient way than SCoR, by ensuring equality in ratings' contribution to the updating process by randomly selecting a pair (user, item) from the ratings' space instead of traversing the list of nodes. Additionally, it introduces three extra terms in the recommendation of a rating between a user and item pair, taking into account the estimated average belief (rating) of the total recommender system, the user and the item average beliefs. ER-SCoR has the same benefits of SCoR that include: being a parameter-free method, having resistance to the cold-start problem, as well as annotating the dataset with important annotations, including the identification of both users and items with unique and/or common characteristics in addition to spotting outlier nodes, and exhibits the same linear computational cost of SCoR. The main contribution of this work concerns the significant improvement of SCoR performance and the study of the Synthetic Coordinates mechanism on the RS problem.
This paper is structured as follows: Sect. 2 reviews the related work for RS. The problem formulation is given in Sect. 3. Sect. 4 presents the proposed ER-SCoR framework. Sect. 5 describes the annotations of a dataset based on the proposed system. Sections 6 and 13 present the experimental results and our conclusions, respectively.
Related Work
As mentioned before, several techniques and approaches have been presented in recent literature, regarding the problem of recommendation. In this Section, we provide a brief overview of the most popular ones. According to the literature consensus, one can divide the various approaches of Recommender Systems into two main categories, namely, {\em Collaborative Filtering} and {\em Content-based}. Collaborative Filtering based approaches generally rely solely on the preferences (e.g., ratings) of users for items, in order to provide the necessary recommendation predictions. In contrast, Content-based Recommender Systems employ additional metadata information, such as attributes/features of both users and items (e.g., music genre, content type, demographic information, etc.) \cite{dtec}.
Collaborative Filtering (CF) approaches pap22,UI2vec,elahi2016survey analyze collective user behavior to infer and deduce each user's preferences and therefore be able to make new predictions. Usually, a number on some preference scale is used to indicate degree of preference. Despite the fact that such approaches usually suffer from cold-start and data sparsity related problems, they benefit from using pre-existing information, which can be provided either implicitly (as users access items) or explicitly (when users evaluate items).
CF Recommender System approaches are usually divided into two major categories, namely, memory-based and model-based pap22. In the first type, the required predictions can be calculated by correlating information, usually by employing a similarity function. This can be done in one of the following ways:
1.
User-to-user: Recommendations rely on similarities between users, often based on their preferences or demographic information.
2.
User-to-item: Recommendations are produced by analyzing the preferences of a user for specific items.
One of the primary techniques used in such systems is collaborative filtering based on memory (or similarity-based) ado12. These algorithms utilize similarity functions to measure the degree of similarity between pairs of users or items, based on historical preferences. Clustering techniques have also been applied in Recommender Systems, either directly tsai12 or as a pre-processing step nil18. For instance, clusters of similar users or items can improve Collaborative Filtering methods by narrowing down the search space to the most relevant candidates.
Model-based Collaborative Filtering (CF) Recommender Systems (RSs) employ various techniques to construct a predictive model that is subsequently used to generate recommendations. These types of approaches usually employ Dimensionality Reduction \cite{pca}, where latent variables are introduced to capture hidden structures underlying user–item interactions. In \cite{UI2vec}, the authors introduce UI2vec, a collaborative filtering model that jointly embeds users and items into a shared latent space using word-embedding techniques, and its enhanced version VUI2vec, which models users and items as Gaussian distributions via variational inference.
In Dimensionality Reduction, each user or item is typically represented as a high-dimensional vector, containing all ratings corresponding to that user or item. However, due to the inherent sparsity of these vectors, since most users rate only a small subset of available items, it becomes challenging to directly identify meaningful correlations between users and items. To address this, Dimensionality Reduction techniques are applied to uncover latent patterns and reduce the complexity of the data. Popular methods include Singular Value Decomposition (SVD) \cite{svd}, Principal Component Analysis (PCA), Probabilistic Latent Semantic Analysis (PLSA), and Latent Dirichlet Allocation (LDA) \cite{plsa}. These approaches transform the original high-dimensional space into a lower-dimensional latent space, where the underlying relationships become more apparent.
The Matrix Factorization method kor09,liu2023recommendation is also a Dimensionality Reduction technique. Both users and items are represented as vectors in a shared latent space, with latent factors inferred from observed rating patterns. Recommendations are generated by identifying items whose latent factors exhibit high similarity to those of a given user.
SCoR \cite{score} utilizes synthetic coordinates, which are assigned to all nodes (users and items), as proposed in kor09, but using the Euclidean distance between a user and an item as opposed to the dot product. Once the system has converged, the aforementioned distance, in the latent space serves as an accurate predictor of the user’s preference for that item. The SCoR framework offers several advantages. It achieves high performance without the need for parameter tuning, and it demonstrates greater robustness to data sparsity compared to alternative approaches. The Vivaldi synthetic network coordinates algorithm \cite{Vivaldi}, which this approach for the RS problem is based on, has proven useful in additional problems to movie recommendation \cite{score,dtec}, such as the identification of malicious profiles in Recommender Systems pan18,pan20a, in personalized video summarization pan20, inn community detection pap14 as well as in interactive image segmentation pan13, resulting in significant increase in performance in comparison with other state-of-the-art methods on publicly available datasets.
In recent years, various approaches have also been proposed in model-based recommendation systems, which employ artificial neural network architectures he2017neural,gao2023survey. Such approaches utilize Convolutional Neural Networks (CNNs) he18, in order to process the output of previous steps (such as the outer product of user to item ratings) to generate a 2D interaction map. This methodology facilitates the model to effectively capture user–item interaction patterns and to learn higher-order correlations.
Recent developments in graph neural networks (GNNs) have utilized embedding propagation to iteratively combine neighborhood embeddings. The information of high-order neighbors can be accessed by the nodes, by stacking propagation layers, outperforming standard methods constrained to first-order neighbors \cite{gao2023survey}.
In \cite{Enriched}, the authors combine an auto-encoder with an enriched matrix concept that adds opposing evaluations of fictional users to those of real users. This led to an increase in the density of the rating matrix, which now incorporates users with more diverse interests and preferences. The work described in \cite{he2017neural} investigates several neural network architectures in the context of collaborative filtering. A general framework is introduced with three distinct implementations: GMF, MLP, and NeuMF, each offering a unique approach to modeling user-item interactions. This work represents a new direction of using deep learning for recommendation, by complementing mainstream shallow models for collaborative filtering.
As mentioned, on the other hand, Content-Based Recommender Systems employ additional metadata information to construct item representations and user profiles, on which recommendation predictions are based pap23. The recommendation process essentially consists of locating items whose features match the user profile attributes, Pasq11, which forms the basis for recommendations. While content-based recommender systems employ mainly textual features to describe the required information of items and user profiles, several hybrid methods have been proposed which employ various information types and/or approaches with the goal of increasing recommendation performance log19.
More recently, the development of Large Language Models (LLMs), such as ChatGPT, DeepSeek and LLaMA, have transformed the domains of Natural Language Processing (NLP) and Artificial Intelligence (AI), allowing for exceptional capabilities in language understanding and generation, along with impressive reasoning and generalization abilities. As a result, recent research has focused on leveraging the power of LLMs to improve RS \cite{zhao2024recommender}. One notable example is Chat-Rec \cite{gao2023chat}, which improves both the accuracy and the explainability of recommendations by integrating ChatGPT into conversational interactions with users. In this approach, ChatGPT refines the candidate item sets originally generated by traditional recommender algorithms, as demonstrated in the context of movie recommendations. Zhang et al. \cite{zhang2023recommendation} employ T5 as an LLM-based approach, allowing users to use natural language in order to specify their explicit preferences, leading to better recommendation accuracy than approaches based merely on user–item interactions.
Problem Formulation
The recommendation prediction problem \cite{score} is formulated hereafter.The input of the problem is a list
of triplets in the form
, where:
1.
belongs to the set
, which consists of distinct identifiers for items that users have rated.
2.
denotes the user rating
for the item
.
The main objective is to estimate the unknown ratings that match the user
's preference for item
, in situations where user
has not provided a rating for
, indicating that
.
In RS research, the most common method to evaluate the accuracy of the predictions provided is to split the original list
into two sets. The first set, referred to as the Training Set (
), contains the “known” user-item ratings that are used to train the recommendation algorithm. The remaining triplets form the Validation Set (
), which is used to evaluate the accuracy of the recommendation algorithm.
To provide accurate predictions, the algorithm calculates a new
value for each existing
value in
. Ideally, the system should converge to such nodes' position such as the values of
match as closely as possible the true values
.This is determined (as is usually the case) with the Root Mean Square Error Metric (RMSE):
RMSE
where
denotes the size (number of triplets) of the
. According to the RMSE definition, a lower calculated RMSE value corresponds to an improved prediction of the RS.
ER-SCoR method
SetKwInOut{Input}{input}
SetKwInOut{Output}{output}
Input{
,
,
,
,
,
,
}
Output{
} \BlankLine
,
ForEach{
}{
random position in
}
ForEach{
}
{
random position in
%arxikopoihsh
}
Repeat{Node positions do not change or
}{
ForEach{
}{
If{(
{\bf and}
) {\bf or} (
{\bf and}
)}{
continue
}
}
If{
}{
Repeat{
and
do not change}{
ForEach{
}{
}
ForEach{
}{
}
}
}
}
ForEach{
}{
}
caption{\label{algo:ER-SCoR} The proposed \textit{ER-SCoR} algorithm.}
algo:ER-SCoR
label{algo:ER-SCoR} The proposed ER-SCoR algorithm.
Fig. 1
label{fig:schema} The schema of the proposed ER-SCoR method.
Fig. 2
label{fig:scor_ex} A synthetic example illustrating the position of nodes (users and items) in
, after the convergence of the system. The smaller the distance between user
and item
, the higher the predicted preference of user
for item
. The predicted preference of user u for item i decreases as the distance between them increases.At each point on the graph, the preference level of the user—positioned at the center is represented through color brightness: light gray denotes high preference (like), while dark gray indicates low preference or dislike.
Here, we present in detail the ER-SCoR recommendation approach for the solution of recommendation prediction problem in linear time O(
), where
denotes the number of given triples in the list
. According to the formulation of the recommendation prediction problem, the input of ER-SCoR is the sets of distinct user identifiers (U) and item identifiers (I), along with the list of triplets formatted as
for both the training set (TS) and the validation set (VS). In addition, the minimum and maximum value of rating (
) are given to constraint the method to provide recommendation values
. The maximum number of iterations of ER-SCoR main loop (
) is also given. Algorithm \ref{algo:ER-SCoR} presents in detail the pseudo-code of the proposed ER-SCoR method. Figure 1 shows the schema of the proposed ER-SCoR method.
In our improved version of SCoRcite{score}, a bipartite graph is created, which consists of user nodes on one side and item nodes on the other.Each
triplet in the Training set (
), is also represented in the graph by a weighted edge connecting the nodes
and
. The basis of this approach is the spring metaphor (see Fig. 2), which was first introduced by the Vivaldi synthetic network coordinate algorithm \cite{Vivaldi}. In this approach a position
,
in
(e.g.
cite{score}) is assigned to each element
,
in the user and the item sets,
and
, respectively. In the original version of SCoR, the distance
between two nodes
is given directly by their Euclidean distance
. In this work, we introduce three extra terms (
,
and
) in the calculation of the distance
, taking into account the estimated average belief (rating) of the total recommender system (
), the user (
) and the item (
) average beliefs.
eq:d
The term
is a term that affects all distances by the average belief (rating) of the total recommender system. The term
adjusts the distance between the user
and any item
, either increasing or decreasing it, to achieve a more accurate recommendation, particularly in cases where synthetic coordinates fail to adequately model the user behavior. For instance, the term
may receive negative values for users who consistently give maximum ratings to most items and especially to those that are far apart in the embedding space. The negative value in term
effectively reduces the distance between user
and those items. Similarly, the term
adjusts the distance between the item
and any user
.
Each edge is assigned a weight equal to the desired distance
between the nodes
and
according to the rating
. We assign a small desired distance value to a pair
with a high rating value
(high preference of user
for item
) and vice versa. Similarly with \cite{score}, we assign the maximum distance (set to 100) to
,the smallest possible rating. In the initial SCoR version \cite{score}, the highest rating was assigned a distance of 0, however, we observed that the zero distance reduces the solution space, resulting in overfitting. Therefore, in order for the highest rating to be assigned a non-zero distance, we include an offset (e.g.
).Given these values, the desired distance
is defined as follows:
eq:dd
where
,
denote the minimum (low preference) and maximum (high preference), respectively. Taking into account Eq. 3 and the values
,
, the recommendation values
are given by Eq. 4.
eq:rr
In the following, we analyze all the steps of the ER-SCoR iterative method.
Firstly, ER-SCoR initializes the values
,
and
to zero and the Synthetic Euclidean Coordinates
,
,
and
to a random position in
(close to zero) (see lines 1-10 of Algorithm \ref{algo:ER-SCoR}). We perform a random permutation on the training set (see lines 13 of Algorithm \ref{algo:ER-SCoR}), so that the edges are traversing in a random way. ER-SCoR iteratively and gradually re-positions all nodes in order for the desired distances of all edges to be satisfied (see lines 14-22 of Algorithm \ref{algo:ER-SCoR}). Ideally, assuming that an item
has been rated by user
with value
, then after convergence, the distance
between the nodes
and
should equal
, as determined by Eq.3.The algorithm iteratively and gradually modifies the positions of each node' (users and items), so that for every known rating
, the Euclidean distance between user
and item
matches the corresponding rating. The algorithm converges when changes in positions more or less stop or the number of iterations exceeds a maximum number. The positions of nodes
and
are updated as follows:
eq:viv_update
where the expression
represents the direction in which node
should be moved and
controls the convergence of the method, by specifying the speed by which node
can move toward its ideal position. It holds that ideally after the system has converged, the distance between the nodes
and
should be
. Upon algorithm convergence, the predicted rating of an item
by a user
consists of a simple calculation of the Euclidean distance between the corresponding nodes. In the special case, where
is equal to the minimum rating
and the distance
is greater than the desired distance
, we skip the synthetic coordinate update process, since the system recommendation
is satisfied (see Eq. 4). Similarly, we skip the synthetic coordinate update process when
is equal to the maximum rating
and the distance
is lower than the desired distance
(see lines 15-17 of Algorithm \ref{algo:ER-SCoR}).
The terms
,
,
of Equation 2 are updated every 50 iterations (see lines 23-33 of Algorithm \ref{algo:ER-SCoR}) taking into account the current positions of nodes. First,
, which corresponds to the average belief of the total recommender system, is calculated by the mean value of the difference
for each edge of the training set. Then
and
are estimated in an iterative process by the corresponding mean values of differences
and
, respectively. The values of
affect the estimation of the values of
and vise versa. So, the update process of
and
also uses the previous values of
and
by the fraction of
for a smooth convergence (see lines 27 and 30 of Algorithm \ref{algo:ER-SCoR}). ER-SCoR terminates when the node positions do not change or the maximum number of iterations
(e.g.
) is reached (see line 34 of Algorithm \ref{algo:ER-SCoR}). Finally, ER-SCoR provides the recommendation values
in the validation set (see lines 35-38 of Algorithm \ref{algo:ER-SCoR}).
In Figure 2, we illustrate a synthetic example that shows the position of the nodes (users and items), after the computation of Synthetic Coordinates. It shows the preferences of the user which is located in the center of the graph. Preferences are visually represented using a gray scale, with light gray indicating like and dark gray indicating dislike.
Dataset Annotations using ER-SCoR
Similarly to SCoR, ER-SCoR not only provides recommendations, but also generates annotations for user and item datasets by analyzing node positions and terms
and
. In our experiments (see Sect. 9), we investigate the ability of ER-SCoR to generate annotations identifying both users of similar tastes and items rated similarly as well as to detect outliers.
Hereafter, we study the relation of the Euclidean distance
and the terms
,
between two user nodes
,
and the maximum absolute difference in their recommendations. Let
be their maximum absolute recommendation difference, according to the ER-SCoR system.
eq:MRD1
By Eq. 4 and the triangle inequality, the supremum (upper bound) of the MRD is:
eq:MRD2
If the distance
between two users
is low, we can assume that these users have similar tastes. Conversely, if a user is placed far from all others in the n-dimensional Euclidean space by ER-SCoR, it suggests that the user exhibits unique or atypical preferences, that is, an outlier. This degree of uniqueness can be quantified by the distance
, which represents the minimum distance between the user
and their nearest neighbor in the entire dataset.
eq:MRD3
The aforementioned approach of users can be extended to the items.
Additionally, ER-SCoR is able to provide annotations for specific users and items by analyzing the values
and
. According to Sect. 4, large positive or large negative values in term
/
indicate users/items that consistently give/receive low or high ratings, respectively. In Sect. 9, we study the behavior of users and items of the five real datasets by analyzing the values
and
.
Experimental Results
Experimental Setup
The following five well-known datasets were used during the experiments presented in the rest of this paper.
The MovieLens dataset (ML-100k) \cite{ml}: The MovieLens dataset, sourced from the GroupLens website \cite{ml}, comprises 100k user ratings from the MovieLens site, with ratings spanning from 1 to 5 across five distinct values.
The MovieLens dataset (ML-1M) (\cite{ml}), sourced from the GroupLens website (\cite{ml}), comprises one million user ratings from the MovieLens platform, with ratings ranging from 1 to 5 (five distinct values).
Jester and Jester2 (\cite{jester,score}): The Jester datasets consist of continuous rating values from several thousand users for 100 jokes, with scores ranging from -10 to 10. To ensure that the RMSE values are comparable with those from other datasets, we adjusted the ratings in the Jester datasets to fall within the range of
, while maintaining the decimal precision.
The Netflix prize dataset (SmallNetflix) (\cite{smallnetflix}): SmallNetflix is a reduced version of the original dataset, available from the GraphLab website (\cite{smallnetflix}). It includes ratings from thousands of users on thousands of movies, using a discrete scale ranging from 1 to 5.
Each dataset was divided into training and validation sets. The first comprises all the ratings used to train the RS and the second whose purpose is to validate and evaluate the performance of the methods.Table \ref{table:data} summarizes the important characteristics for each dataset used, namely the number of users, the number of items (e.g. movies, jokes), the number of user ratings, the average number of ratings per user (ratings/user) and the density of the dataset. The density is determined by the fraction of existing ratings relative to all possible ratings, expressed as
.
In Figs. 3, 4 and 5, the Probability Mass Function (PMF) of discrete rating values is shown for: ML-100k, ML-1M, and SmallNetflix datasets, respectively. In Figs. 6 and 7, the Probability Density Function (PDF) of continuous rating values is shown for Jester and Jester2 datasets, respectively. In all datasets, higher ratings occur more frequently than lower ones.
Figure 17 depicts the PDFs of user and item degree values for ML-100k, ML-1M, Jester, Jester2 and SmallNetflix datasets. In all datasets, the range of item degrees is greater than the corresponding range of user degrees. Probability density functions (PDFs) are estimated by employing a normal kernel function and are evaluated at evenly distributed points spanning the entire range of ratings or degrees.
begin{table}[]\begin{tabular}{|l|r|r|r|r|r|r|}\hlineDataset & \multicolumn{1}{l|}{Users} & \multicolumn{1}{l|}{Items} & \multicolumn{1}{l|}{Ratings} & \multicolumn{1}{l|}{TS Ratings} & \multicolumn{1}{l|}{
} & \multicolumn{1}{l|}{TS Density} \\ \hlineML-100k & 943 & 1682 & 100000 & 80000 & 84.84 & 5.04% \\ \hlineML-1M & 6040 & 3952 & 1000209 & 800167 & 132.48 & 3.35% \\ \hlineJester & 23500 & 100 & 1708993 & 1367194 & 58.18 & 58.18% \\ \hlineJester2 & 24938 & 100 & 616912 & 493529 & 19.79 & 19.79% \\ \hlineSmallNetflix & 93705 & 3561 & 3843340 & 3074672 & 32.81 & 0.92% \\ \hline\end{tabular}\caption{\label{table:data} Statistics for the datasets used in the experiments.}\end{table}
Fig. 8
label{fig:hist} The Probability Mass Functions (PMFs) of discrete rating values for (a) ML-100k, (b) ML-1M and (c) SmallNetflix datasets. PDFs of continues rating values for (d) Jester and (e) Jester2 datasets.
In \cite{score}, SCoR system was tested along with seven state-of-the-art similar RSs (ALS \cite{als}, ALS_COORD \cite{als_coord}, BIASSGD \cite{biassgd}, BIASSGD2 \cite{biassgd}, RBM \cite{rbm}, SGD \cite{sgd}, SVDPP \cite{biassgd}, P_MEAN [Ekstrand:2011:CFR:2185827.2185828] and USER-USER algorithm \cite{Sarwar}), resulting the highest performance under any of the following datasets: (smallnetflix) \cite{smallnetflix}, the MovieLens dataset (ml) \cite{ml}, jester and jester2 \cite{jester}. This study assesses the performance of ER-SCoR in comparison to SCoR and two more recent state-of-the-art recommendation approaches \cite{Enriched, UI2vec}, introduced after SCoR.
Our algorithm does not require parametrization, with the sole exception of the number ofdimensions of the Euclidean space. This value was set to
in all experiments, as proposed in the initial version of SCoRcite{score}. The constant
(see Sect. 4) is set to 20 forall datasets. A sensitivity analysis of ER-SCoR under different values of
is presented in Sect. 11.
Performance Evaluation
In this Section, the performance of the ER-SCoR over the used datasets is studied.
Figure 24 depicts the evolution of RMSE during training process of ER-SCoR for different values of ratings on training (left column) and validation (right column) set of ML-100k, ML-1M and SmallNetflix datasets. As it was expected, the
RMSE values are lower on the training set compared to the validation set, while maintaining the same relative order with respect to the rating values. The lowest RMSE values are observed for rating three that is the middle rating and rating four that has the highest frequency in the datasets (see Fig. 8). The highest RMSE values is observed for the rating one (extreme value), since it has the lowest frequency in the datasets. Concerning the convergence of RMSE sequence, due to the periodic update process of
,
and
, a slight discontinuity can be observed, particularly at the fiftieth iteration, when the first update occurs. Beyond this point, the system exhibits smooth and consistent convergence across all datasets and rating values.
Figure 35 depicts the evolution of RMSE during training process of ER-SCoR for low (bottom
) and high (top
) degree users and items on training (left column) and validation (right column) datasets. Due to overfitting, on the training set the lowest RMSE values are observed for low-degree items and users. In contrast, in the test set, the highest RMSE values are associated with items and users of low-degree. Concerning the convergence of RMSE sequence, the system almost always converges smoothly. The periodic update of
,
and
introduces a slight discontinuity, especially noticeable in the SmallNetflix dataset during the first update iteration.
Fig. 17
label{fig:histDeg} PDFs of user and item degree values for (a) ML-100k, (b) ML-1M, (c)-(d) Jester, (e)-(f) Jester2, and (g)-(h) SmallNetflix datasets.
Fig. 24
label{fig:RMSE_Rats} The evolution of RMSE during training process of ER-SCoR for different values of ratings on training (left column) and validation (right column) datasets.
Fig. 35
label{fig:RMSE_Deg} The evolution of RMSE during training process of ER-SCoR for low and high degree users and items on training (left column) and validation (right column) datasets.
ER-SCoR annotations
The ER-SCoR based annotations for the ML-100k, ML-1M, Jester, Jester2, and SmallNetflix datasets are given hereafter.
Figure 41 illustrates the cumulative distribution functions (CDFs) of
of ER-SCoR estimated on ML-100k, ML-1M and SmallNetflix. The CDFs for items and users are illustrated with blue and curves, respectively.According to ER-SCoR method, the distance between users and items is generally in
. We can set a threshold of
on the maximum value of the distance (which corresponds to the maximum rating difference by one in a five-degree rating system) to identify similar users and items. Furthermore, a threshold of
on the maximum distance value (which corresponds to the maximum rating difference by three in a five-degree rating system) has been used to detect outliers. Based on these rules and the CDFs of Fig. 41, Table \ref{table:Dmin} depicts the detection rates of similar users/items and outliers by the ER-SCoR system. The results show that the SmallNetflix dataset has the highest percentage of similar users (
), which is expected given that it contains the largest number of users. In contrast, the ML-100k dataset, which has the fewest users, exhibits the lowest percentage of similar users (
). The ML-100k and ML-1M datasets exhibit the highest percentage of similar items (about
), which is expected given their large number of items. In contrast, the two Jester datasets, which contain only 100 items, result in zero similar items. Regarding outliers, the Jester dataset exhibits the highest percentage of user outliers, at
, while the SmallNetflix dataset shows
item outliers.
Figure 47 depicts the values of
and
sorted in ascending order on ML-100k, ML-1M, Jester, Jester2 and SmallNetflix datasets. Regarding users, in all datasets, there are a few users with highly negative values (less than -10) on
, indicating that they consistently give high ratings. In the SmallNetflix dataset, a few users exhibit highly positive values (greater than 10) on
, meaning they consistently give low ratings.As for items, in the ML-100k and ML-1M datasets, a small number of items have high positive values (greater than 10) on
, meaning they consistently receive low ratings. Table \ref{table:DUI} shows the average values and the standard deviation of
,
and the
per dataset.
is positive in all cases, indicating that the average belief across the datasets is that users tend to give, and items tend to receive, high ratings, an observation that is confirmed by Fig. 8. The relatively small mean values compared to the standard deviations suggest the existence of users and items with divergent beliefs in every dataset.
Fig. 41
label{fig:Dmin} The Cumulative Distribution Function (CDF) of
for ER-SCoR on ML-100k, ML-1M, Jester, Jester2 and SmallNetflix datasets.
begin{table}[]\begin{tabular}{|l|rr|rr|}\hlineDataset & \multicolumn{2}{c|}{Similar} & \multicolumn{2}{c|}{Outliers} \\ \hline & \multicolumn{1}{l|}{Users} & \multicolumn{1}{l|}{Items} & \multicolumn{1}{l|}{Users} & \multicolumn{1}{l|}{Items} \\ \hlineML-100k & \multicolumn{1}{r|}{2.76%} & 14.86% & \multicolumn{1}{r|}{0.32%} & 1.25% \\ \hlineML-1M & \multicolumn{1}{r|}{8.49%} & 15.56% & \multicolumn{1}{r|}{0.12%} & 0.84% \\ \hlineJester & \multicolumn{1}{r|}{8.94%} & 0.00% & \multicolumn{1}{r|}{0.63%} & 0.00% \\ \hlineJester2 & \multicolumn{1}{r|}{7.08%} & 0.00% & \multicolumn{1}{r|}{0.08%} & 1.00% \\ \hlineSmallNetflix & \multicolumn{1}{r|}{34.44%} & 7.39% & \multicolumn{1}{r|}{0.02%} & 2.19% \\ \hline\end{tabular}\caption{\label{table:Dmin} The detection rates of similar users/items and outliers by the \textit{ER-SCoR} system.}\end{table}
Fig. 47
label{fig:DUI} The values of
and
sorted in ascending order on ML-100k, ML-1M, Jester, Jester2 and SmallNetflix datasets.
begin{table}[]\begin{tabular}{|l|rr|rr|r|}\hlineDataset & \multicolumn{2}{c|}{Average} & \multicolumn{2}{c|}{Std Dev} & \multicolumn{1}{l|}{
ML-100k & \multicolumn{1}{r|}{-0.14} & 0.37 & \multicolumn{1}{r|}{1.98} & 1.40 & -1.10 \\ \hlineML-1M & \multicolumn{1}{r|}{0.02} & 0.33 & \multicolumn{1}{r|}{1.50} & 1.06 & -0.84 \\ \hlineJester & \multicolumn{1}{r|}{0.04} & 0.09 & \multicolumn{1}{r|}{1.15} & 0.81 & -0.90 \\ \hlineJester2 & \multicolumn{1}{r|}{-0.04} & 0.53 & \multicolumn{1}{r|}{1.44} & 1.46 & -1.02 \\ \hlineSmallNetflix & \multicolumn{1}{r|}{-0.15} & 0.53 & \multicolumn{1}{r|}{2.04} & 1.21 & -0.96 \\ \hline\end{tabular}\caption{\label{table:DUI} The average values and the standard deviation (Std Dev) of
,
and the
on \textit{ML-100k}, \textit{ML-1M}, \textit{Jester}, \textit{Jester2} and \textit{SmallNetflix} datasets.}\end{table}
ER-SCoR convergence
Fig. 50
label{fig:AbsError} The evolution of absolute validation error of RMSE during training process of (a) SCoR and (b) ER-SCoR on the ML-100k, ML-1M,Jester, Jester2 and SmallNetflix datasets.
In this Section, we study the convergence of ER-SCoR during training process compared with the SCoR method. Figure 50 shows the evolution of absolute validation error (
) of RMSE during training process of SCoR and ER-SCoR on the ML-100k, ML-1M,Jester, Jester2 and SmallNetflix datasets, defined as follows:
eq:e_n
where the first term denotes the converged RMSE on the validation set and the
denote the RMSE at validation set at
iteration. According to this experiment, ER-SCoR converges in at most 200 iterations across all datasets, whereas SCoR requires up to 5000 iterations to converge. The difference of approximately 25 times in speed convergence can be mainly explained by the new position update mechanism of ER-SCoR that ensures equality in rating contribution and updates the position of both nodes of the selected edge (bidirectional position update), while SCoR updates only the position of the selected node. Additionally, SCoR promotes balanced participation among all nodes, assigning greater influence to ratings from low-degree nodes compared to those from high-degree ones.
ER-SCoR sensitivity test
Fig. 51
label{fig:sens_test} The RMSE of ER-SCoR-PU,
ER-SCoR on ML-100k dataset for different values of constant
(sensitivity test).
In this Section, the sensitivity test of ER-SCoR and two variants of the proposed method ER-SCoR-0,
is studied. Figure 51 shows a sensitivity test of the proposed method
ER-SCoR and ER-SCoR-0,
in the ML-100k dataset. According to the sensitivity test, all the methods results similar RMSE for any
, showing that the methods are not sensitive to the selection of constant
. In this work, the constant
is selected equal to 20 under any dataset. Similar results are obtained for the remaining datasets.
Comparisons with other Recommender Systems
begin{table}[]\begin{tabular}{|l|r|r|r|r|r|r|}\hlineSystem\textbackslash{}Dataset & \multicolumn{1}{l|}{ML-100k} & \multicolumn{1}{l|}{ML-1M} & \multicolumn{1}{l|}{Jester} & \multicolumn{1}{l|}{Jester2} & \multicolumn{1}{l|}{SmallNetflix} & \multicolumn{1}{l|}{Average} \\ \hlineER-SCoR-PU & 0.910 & 0.853 & 0.816 & 0.885 & 0.885 & 0.870 \\ \hline
ER-SCoR & 0.900 & 0.848 & 0.812 & 0.878 & 0.876 & 0.863 \\ \hlineSCoRcite{score} & 0.933 & 0.894 & 0.843 & 0.893 & 0.921 & 0.897 \\ \hlineEnriched_AE \cite{Enriched} & 0.964 & 0.921 & 0.852 & 0.918 & 0.943 & 0.920 \\ \hlineUI2vec \cite{UI2vec} & 0.977 & 0.924 & 0.854 & 0.940 & 0.950 & 0.929 \\ \hline\end{tabular}\caption{\label{table:res} The \textit{RMSE} values for \textit{ER-SCoR}, two variations of \textit{ER-SCoR} (\textit{ER-SCoR-PU}, \textit{ER-SCoR-PU-D
}) three recommender systems from literature (\textit{SCoR \}\cite{score}, Enriched_AE \cite{Enriched}and \textit{UI2vec \cite{UI2vec}}) on the \textit{ML-100k}, \textit{ML-1M},\textit{Jester}, \textit{Jester2} and \textit{SmallNetflix} datasets. The last column depicts the average \textit{RMSE} value of each recommender system computed over the five datasets. Top scores per dataset are highlighted in bold. }\end{table}
Table \ref{table:res} depicts the RMSE values for ER-SCoR, two variations of ER-SCoR (ER-SCoR-PU,
SCoR cite{score}, Enriched_AE \cite{Enriched}/TEXT> and UI2vec \cite{UI2vec}) on the ML-100k, ML-1M,Jester, Jester2 and SmallNetflix datasets. The last column shows the average RMSE value of each recommender system computed on the five datasets. Under any dataset ER-SCoR clearly outperforms all the methods from literature and slightly outperforms the two variations of ER-SCoR (ER-SCoR-PU,
ER-SCoR-PU due to the extra term
On average, ER-SCoR yields
and
lower RMSE than the two variations
ER-SCoR-PU, respectively. Furthermore, ER-SCoR yields
,
and
lower average RMSE than the three recommender systems from literature SCoR, Enriched_AE and UI2vec, respectively. The best results of ER-SCoR are obtained under ML-1M dataset, where ER-SCoR yields
,
and
lower RMSE than the SCoR, Enriched_AE and UI2vec, respectively. According to Table \ref{table:res}, among the baseline methods, SCoR performs best, followed by Enriched_AE, which in turn outperforms UI2vec.
Figure 55 shows the RMSE of SCoR, ER-SCoR-PU,
ER-SCoR in the ML-100k, ML-1M and SmallNetflix datasets with different ratings values. The outperformance of ER-SCoR and the variations of ER-SCoR (ER-SCoR-PU,
SCoR is more clear for the extreme values, especially for
. The primary reason for the difference in performance is the new position update mechanism of ER-SCoR that employs the offset
to mitigate the overfitting issues of SCoR (see Sect. 4) and to the equal impact on used ratings. Furthermore, the equality in rating contribution results in the improved performance of ER-SCoR, as the system tends to produce lower errors on the ratings most frequently observed.
Fig. 55
label{fig:RMSE_Rat} The RMSE of SCoR, ER-SCoR-PU,
ER-SCoR on ML-100k, ML-1M and SmallNetflix datasets under different values of ratings.
Conclusions
In this work, we introduced ER-SCoR, an Equal Ratings impact-based Recommender System utilizing Synthetic Coordinates. ER-SCoR enhances the original SCoR model by ensuring equal contribution of all ratings during coordinate updates and by integrating three additional belief-based terms (global, user-specific, and item-specific) into the recommendation process. These modifications significantly improve the performance, convergence speed and stability of the system.
Extensive experiments conducted on five widely-used datasets (ML-100k, ML-1M, Jester, Jester2, and SmallNetflix) demonstrated that ER-SCoR consistently outperforms both the original SCoR and two state-of-the-art methods, including deep learning and matrix factorization approaches. In particular, ER-SCoR achieves faster convergence, up to 25 times faster than SCoR, while consistently achieving lower RMSE values across all evaluated datasets. The method maintains all advantages of the SCoR framework, such as parameter-free operation, robustness to cold-start problems, and linear computational complexity. Beyond prediction accuracy, ER-SCoR offers interpretative value through the use of synthetic coordinates and belief terms. It enables intuitive dataset annotations and the detection of user/item outliers, thereby offering insights not only into what is recommended, but also why and to whom-a property of increasing importance in transparent and explainable AI systems. Conclusively, ER-SCoR represents a robust and efficient framework in the field of recommender systems, offering strong theoretical foundations, empirical superiority, and extensibility towards next-generation intelligent recommendation frameworks.
Future work will focus on incorporating side information and extending the model to support dynamic or sequential recommendation scenarios. Additionally, adapting the coordinate update mechanism for distributed environments could enhance scalability and computational efficiency.
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C.P. wrote the main manuscript text and prepared figures 1-11. All authors reviewed the manuscript.References:
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Acknowledgements
This work was supported by Project “Enhancing the Greek Safer Internet Center SaferInternet4Kids: Awareness, Helpline, Hotline”, DIGITAL EUROPE, 2024–2026.
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