To predict recommendations to the users based on past behavior and their preferences, a technique in recommender systems named collaborative filtering is used [1]. The main objective is to suggest items or content to the users by considering their interactions or resemblances with other users. Due to the continuous growth in data availability and the interconnected nature of diverse systems, these techniques hold immense importance in revealing patterns, relationships, and significant insights [2].

Matrix factorization (MF) is a technique utilized in collaborative filtering to decompose a matrix of user-item ratings into lower-rank matrices capturing the latent factors underlying the data ^[3][4][3,4]. The user-item rating matrix serves as a representation of user ratings assigned to different items [5]. The matrix is frequently sparse as users typically only rate a small subset of items. MF techniques strive to complete the missing entries in the matrix by decomposing it into lower-rank matrices, one capturing the user’s underlying preferences and the other reflecting the item’s latent characteristics [6]. The latent representations of users along with items can be used to estimate future ratings or calculate missing ratings once the matrix has been factorized.

In network analysis, community detection stands as a fundamental task, endeavoring to identify cohesive subsets of nodes, referred to as communities or modules within a network [7]. These communities in networks signify groups of nodes that display stronger interconnections amongst themselves compared to connections with nodes outside the community [8]. These communities provide insightful information on the structure, organization, and dynamics of complex systems ^[9][10][9,10]. The applications of community detection across various domains, such as social network analysis, biological networks, online forums, and recommendation systems, have garnered substantial attention [11]. Through the identification of communities, we gain insights into the underlying interaction patterns, discover influential groups, and develop a deeper comprehension of the network’s functionality [12].

Matrix factorization techniques have been widely applied across diverse domains, demonstrating their versatility and effectiveness. Some of the notable applications of the matrix factorization techniques include

In essence, the applications of MF techniques have a broad scope of transforming the methods through which we analyze, comprehend, and leverage complex data [23]. The ongoing evolution of these techniques holds the potential to unlock fresh possibilities and propel advancements in numerous domains. It ultimately benefits individuals, organizations, and society at large.

2. MaLitrix Factorization in Recommendation Systemserature Review

In recent years, the MF method has garnered significant attention as a widely adopted and successful method for rating prediction in recommendation systems. The Netflix Prize competition, which was started in 2006, is one early piece of work noteworthy in relation to MF in recommender systems ^[24][25]. The fundamental matrix factorization model creates user and item latent feature matrices from the rating matrix, enhancing the accuracy of rating predictions through the understanding of possible connections between users and items. As the information interconnection era has emerged, the basic MF model no longer satisfies the demands of recommender systems. It thus leads to the emergence of numerous variants of this model. Excluding all the nonnegative entries in latent features, a model nonnegative matrix factorization (NMF) was initiated by Paatero and Tapper in 1994 that improves the accuracy of the model ^[25][26]. Mnih et al. in the year 2007 proposed probabilistic matrix factorization (PMF), which utilizes probabilistic modeling to effectively capture uncertainties in user-item ratings, resulting in recommendations that are more reliable and precise ^[26][27]. The significance of the singular value decomposition (SVD) technique was introduced by Mastorakis in 1857 and researchers started to apply SVD to the recommendation domain from 2006 ^[27][28]. The method has the ability to perform robust MF, enabling the identification of hidden features and effective dimensionality reduction in data analysis. By integrating explicit and implicit feedback to enhance recommendation accuracy and personalization, Koren et al. in the year 2008 introduced an advanced singular value decomposition (SVD++) method ^[28][29]. Through the incorporation of user-item ratings and implicit feedback, SVD++ boosts the performance of the recommender systems, enabling better capture of user preferences and more effective recommendation generation. In 2015, Shi et al. introduced a method named pairwisely constrained nonnegative symmetric matrix factorization (PCSNMF) that incorporates the symmetric community structures found in undirected networks but also leverages pairwise constraints derived from ground-truth group information ^[29][30]. In 2017, deep matrix factorization (DMF) was introduced by Xiangnan et al. and the deep learning with MF ^[30][31] emerges. This method enables the discovery of intricate patterns and the extraction of complex features from large-scale datasets. Kipf and Welling in the year 2018 combined the power of MF and graph convolutional neural networks to capture both collaborative filtering patterns and graph structures in recommendation systems ^[31][32]. By incorporating the graph information, the recommendation accuracy is improved by leveraging the connectivity and relationships among users and items. In 2019, factorized asymmetric nonnegative matrix factorization (FANMF) was introduced by Tosyali et al. by considering the asymmetric relationships between users and items ^[32][33]. It led to enhanced recommendation quality, a deeper understanding of user preferences, and ultimately provided personalized user experiences in various data analysis tasks.