# Two-Layer Matrix Factorization and Multi-Layer Perceptron for Online Service Recommendation
## 2 Related Work  
Capturing linear relationships between users and items is important to the performance of recommender system models [10; 11; 12]. Traditional matrix factorization techniques can capture the first-order linear relationships using self interaction functions. However, in practical situations, there often exits second-order or higher-order linear relationships between users and items. [PERSON] et al. [13] proposed a model using Factorization Machine (FM) to capture the first-order and second-order linear relationships between users and items; thus, the model's ability to capture linear relationships has been improved. [PERSON] et al. [14] found that different feature domains of users and items exhibit heterogeneity when interacting, which hinders the linear capture ability of the model. Therefore they proposed the Field-Aware Factorization Machines (FFMs) model, which stores several embeddings for one feature field. The corresponding embeddings are selected accord ing to interacting objects in the feature domain, which improves the effect of the model. However, it is not enough to rely solely on linear features to output recommender system feedback. The categories of data input to the recommender system are diverse and discrete, forming a huge sparse matrix [3]. New methods are needed to jointly capture the linear and nonlinear features of the data space. In recent years, because of powerful nonlinear fitting and feature representation capabilities, deep learning has been gradually introduced into recommender systems [15; 16]. It is not enough to rely only on nonlinear relationship mining, so many scholars have tried to combine deep learning with linear models. In order to enable the model to capture high-order linear interactions, [PERSON] et al. [3] proposed the Deep Cross Network (DCN) model, which uses a multi-layer perceptron to capture the higher-order implicit interactions between features. The higher-order linear interactions are captured by repeatedly sending the original embedding vectors into the cross network for calculation, which can effectively reduce the model parameters and achieve a high accuracy rate. [PERSON] et al. [2] proposed the Wide and Deep model, which uses a multi-layer perceptron to learn deeper user information to recommend more diverse items. However, due to limited interactions between users and items, large amounts of irrelevant items were recommended by the model. In that case, a linear model was introduced to capture the linear relationships, so that more related items could be recommended, and the overall accuracy of the model improved. Since the pure DNN network cannot capture the relationships between different feature fields, [PERSON] et al. [17] proposed the Product-base Neural Network (PNN) model, which is the first model using the embedding technology to convert the high-dimensional input into a low-dimensional dense embedding, and then input it into the product layer to capture the interactions between different features of the input data, extract their linear and nonlinear interactions, and finally input them into the subsequent multi-layer perceptron for further extraction of the higher-order interactions. To summarize, since deep learning contains a large number of parameters, a large amount of training is required to learn any interaction function, which makes it difficult for models to capture linear relationships between users and items with only deep learning. Consequently, overfitting easily occur in the low-frequency data region. When the user's historical evaluation information is not enough, it is difficult for the recommendation system to accurately grasp the user's preferences. At the same time, although the above-mentioned DCN, DNN, and other models introduce linear models to make up for the insufficiency of deep learning, they cannot dynamically adjust the dimensions of their embedding vectors. Using high-dimensional embedding in scenarios with small data volumes is still prone to overfitting, which hinders the capture of linear relationships.