## 1 Introduction  
Recommendation systems are widely used in real-life recommendation tasks as an effective method to solve the issue of information overload. Collaborative filtering algorithm is the most classic and commonly used in the field of recommendation system. The basic idea of collaborative filtering algorithm is to recommend items to users based on their previous preferences and the choices of other users with similar interests, that is, based on users' historical behavioral data to discover users' preferences and to predict the items that users may like. It does not rely on any items' additional information (e.g., their own characteristics) or any users' additional information (e.g., age, gender, etc.). Currently, the most widely used collaborative filtering algorithm is the neighborhood-based method, which includes two main types of algorithms. One type is the user-based collaborative filtering algorithm (UserCF), which recommends products to users according to others has the similar interests as them, the other is the item-based collaborative filtering algorithm (ItemCF), which recommends items which are similar to their previously preferred items. Additionally, recommender systems usually need to collect users' personal information and interaction records to train models and perform high-quality product recommendations [1, 2]. Traditional recommender systems assume that platform parties and users are completely trustworthy with each other. However, the reality is often not as ideal as we think, there often exists the risk of privacy leakage in real scenarios, and this risk exists in multiple ways, including between users and platforms, between users and users, and between platforms and platforms. This leads to the risk of leakage of users' private information. Collaborative filtering algorithms are widely applied in various fields; however, these algorithms extensively use users' historical information in the process of preference calculation, which may lead to the risk of privacy leakage [3; 4; 5]. Therefore, privacy protection in recommendation algorithms is of great concern to users. To address the issue of privacy leakage during recommendation, many studies have tried to bring the differential privacy-preserving strategy into collaborative filtering algorithms [6]. Some typical algorithms include one designed by [PERSON] et al. [7], which is an item-based differential privacy recommendation algorithm and a user-based differential privacy recommendation algorithm, which can effectively solve the issue of privacy leakage of collaborative filtering algorithms, but due to its neglect of the sensitivity of user rating data, a large amount of noise data is introduced, resulting in unsatisfactory algorithm efficiency, and at the same time, the user-based strategy and the project-based strategy are independent to each other, so advantages of algorithms are not fully released. [PERSON] et al. [8] combined the Bhattacharyya similarity with the K-medoids clustering to improve the accuracy of the similarity measure of the differential privacy-based collaborative filtering algorithms. Meanwhile, the differential privacy mechanism also introduces a lot of noise data that impairs the performance of the recommendation system. How to trade-off between algorithmic performance and the privacy security has become one of the hot spots in the field of differential privacy-preserving recommendation algorithms. To better protect users' privacy security, many scholars have tried to introduce the method of data reconstruction, which is an evaluation algorithm based on the probabilistic perturbation mechanism. Traditional evaluation algorithms can only reconstruct the distribution of a single vector; thus, [PERSON] et al. [9] proposed a joint cardinality estimation method to reconstruct the joint distribution of multiple vectors. Then, [PERSON] et al.