# Weighted Matrix Factorization Recommendation Model Incorporating Social Trust
## 1 Introduction  
With the development of the Internet, information in the network has grown explosively, and the problem users face has shifted from a shortage of information in the past to how to efficiently filter and obtain valuable information in the current environment of information overload. To address this, recommendation systems [1] have emerged. Traditional recommendation models include content-based recommendation models [2], collaborative filtering recommendation models (user-based [3] and item-based collaborative filtering [4]), and matrix factorization-based recommendation models [5]. The core idea of content-based recommendation models is to use the content attributes of items to recommend items similar to the user's historical preferences. For example, if a user has liked many science fiction movies in the past, a content-based recommendation model might recommend more science fiction movies to that user. In content-based recommendation models, item features are typically constructed, and then the user's preference model is learned based on past user behaviors like ratings or purchases. The recommendations are made according to this model. Collaborative filtering models primarily recommend based on the similarity between users or items. There are two main types: user-based collaborative filtering and item-based collaborative filtering. In user-based collaborative filtering, the model finds other users with similar preferences to the target user by calculating the similarity between users. The model predicts the target user's ratings for unrated items based on these users' rating data and makes recommendations accordingly. The rating prediction is usually achieved by a weighted average of these similar users' ratings, where the weights are the similaritiesbetween users. Compared to the user-based approach, item-based collaborative filtering focuses on the similarity between items. The model calculates the similarity between various items based on the ratings given by users to these items. If a group of users gives similar ratings to two different items, those items are considered similar. When a user rates an item highly, the model recommends other similar items based on the assumption that the user may have similar preferences for similar items. Both user-based and item-based collaborative filtering face the cold start problem [6]. When new users join the model, it is hard to find other users similar to them due to the lack of their historical ratings, making it difficult to provide effective recommendations; new items cannot be accurately compared and matched with existing items by the model until they receive enough user ratings; hence, they are hard to be recommended to users. The core idea of recommendation models based on matrix factorization is to decompose a large user-item rating matrix into two smaller matrices to reveal the latent relationships between users and items. The model decomposes the user-item rating matrix into a user feature matrix and an item feature matrix, where the rows and columns of each matrix represent the latent features of users and items, respectively. These features, trained through algorithms, reflect the preferences of users and the attributes of items. The key to matrix factorization lies in training the model by minimizing the difference between the user rating matrix and the product of the decomposed user feature matrix and item feature matrix. Once these latent features are obtained, they can be used to predict the likely ratings of a user for unrated items through their interactions, thereby generating personalized recommendations. This method is suitable for processing large-scale datasets and can alleviate the cold start problem to some extent, but it still faces the issue of limited recommendation performance due to missing data in the user-item rating matrix. With the rapid development of social networks, social recommendation has become an important research direction [7].