# Two-Layer Matrix Factorization and Multi-Layer Perceptron for Online Service Recommendation
## 1 Introduction
the overall effect of the model. Since capturing higher-order linear interactions requires a large number of parameters, [PERSON] et al., inspired by residual network [9], proposed a Deep and Cross Network (DCN) model [3], in which a multi-layer perceptron was used to capture implicit higher-order interactions between features. Meanwhile, they used the cross network to capture higher-order linear interactions, improving the linear capture capability of the model. [PERSON] et al. [2] proposed the Wide and Deep model, which applies the multi-layer perceptron to learn deeper user-item information to recommend more diverse items and uses linear functions to capture the linear relationships between users and items, so as to recommend more related items. However, these works were all relying on fixed embedding vectors. They did not incorporate multi-dimensional embedding into the model and still used high-dimensional embeddings for low-frequency data. It would make the model prone to overfitting for low-frequency users, thus preventing the model from capturing linear user-item relationships. In summary, in order to better solve the phenomenon of low frequency data overfitting, to allow the deep learning model to capture linear relationships, and to address defects such as being incapable of integrating the multi-dimensional embedding, this paper proposes the Two-layer Matrix Factorization and Multi-layer Perception Neural Network (TMMNN). In our model, the dimension of the embedding vectors is adjusted according to the data frequency in the adjustment layer to form multi-dimensional embedding vectors. Matrix factorization is introduced as a linear model that linearly combines the latent features of the input data. The neural network of the multilayer perceptron is used to perform in-depth learning of low-dimensional dense embedding vectors and to generalize the unknown features in a low-dimensional way to improve the generalization characteristics of the model. Hence, this model can receive embeddings of different dimensions and capture nonlinear interactions between users and items through the powerful nonlinear fitting ability of the multi-layer perceptron. Moreover, the model can use the linear function of matrix factorization to capture linear user-item relationships. Compared with the ESAPN model, the advantage of our model lies in the introduction of a linear capture technique--matrix factorization, which is applied especially to capture linear relationships. Our model combines matrix factorization with the dynamic dimension adjustment technology of an embedding vector. Thus, the problem of overfitting in the low-frequency data area can be further solved, and the model's recommendation efficiency with middle- and high-frequency data can be improved at the same time. Compared with models such as DCN and Wide and Deep, our model has some advantages as well. In our model, the embedding dynamic adjustment technology is introduced, which can dynamically process the input data with changing frequency. This technology is conducive to the capture of linear relationships, and can optimize the memory while improving the recommendation performance of the model for low-frequency data. In this paper, Section 2 reviews the related research work; Section 3 details the structure and implementation of the TMMNN model; Section 4 discusses the experimental design of the TMMNN model on two evaluation indicators, together with the analysis and discussion of the experimental results; and Section 5 gives the conclusions and future work.