# Two-Layer Matrix Factorization and Multi-Layer Perceptron for Online Service Recommendation
## 1 Introduction  
In recent years, the scale of the Internet has continued to expand, and the online service industry has developed rapidly. Users are often faced with a massive number of services in the network. Thus, how to accurately push services based on users' historical interaction information with items has become a hot research topic. The development of service recommendation is crucial to improving user experience. Lately, many researchers have established a variety of models to capture linear and nonlinear relationships between users and items to predict user preferences for network services. By means of the powerful nonlinear fitting and feature learning capabilities, deep learning could capture the implicit interactions between users and services in recommendation tasks to improve recommendation accuracy [1]. In the meantime, the Wide and Deep model [2] introduced a wide linear model when training deep neural networks. It is difficult for the neural network to learn the sparse and high-order query matrix at the bottom, resulting in poor feedback from the recommendation system in the occurrence of this special situation [2,3]. Therefore, the linear model can be used for auxiliary processing to capture linear relationships and to improve the memory ability of the overall model. The Deep and Cross Network (DCN) [3] obtains the weight of the cross term by product, and the feature parameters are linearly combined to filter meaningless prediction information. Both models add a linear module to capture linear relationships between users and items, which improves the prediction accuracy. Their success indicates that the capture of linear relationships of the model also has an important impact on the recommendation accuracy. However, although the recommendation accuracy is relatively high when dealing with large datasets, the overfitting phenomenon happens when deep learning models are capturing linear relationships between users and items with reduced dataset sizes [4,5]. Figure 1 shows the prediction accuracy of the Embedding Size Adjustment Policy Network (ESAPN) [4] for users with different frequencies, where the ordinate represents the model's prediction accuracy and the abscissa represents the user's frequency. This model is selected as the comparison target in the following section, so the experimental acquisition process of the data in the figure will be introduced later. The model dynamically adjusts the embedding size for user-item feature pairs according to frequency, trying to address the problem of overfitting caused by high-dimensional embedding when dealing with low-frequency datasets. However, it can be found in the figure that a low frequency of user-item interactions corresponds to a lower prediction accuracy in the model, and there still exists a certain degree of overfitting. The reason for this phenomenon is that the deep learning model contains a large number of parameters and requires large amounts of data for training to learn a linear function [6,7]. Low user-item frequency makes it difficult for the model to capture linear relationships between users and items and in turn affects the prediction accuracy. It further illustrates that the linear relationship of the model has a significant impact on the recommendation accuracy. Therefore, in addition to the method of dynamically changing the embedding dimension proposed by the ESAPN model, the overfitting phenomenon can be further alleviated by strengthening the capture of linear relationships, thus improving the prediction accuracy of the model as a whole. Linear capture technology can improve the performance of deep learning models in capturing linear relationships. [PERSON] et al. [8] designed a neural network model based on a Factorization Machine (FM), in which a multi-layer perceptron was introduced to capture the implicit higher-order interactions between features. In this model, FM was used to capture the second-order linear interactions between feature d  
Figure 1: Prediction accuracy of users with different frequencies.