## 2 Related Work  
Before deep learning was widely developed and applied to recommendation systems, the core algorithm of the recommendation system was always collaborative filtering. The classical application of the collaborative filtering algorithm can be traced back to the mail filtering system of the Xerox Research Center in 1992 [14]. However, its development was dependent on Amazon [15] in 2003, making it a well-known classic model. Until now, all major models have been influenced by it. Later, in the 2006 Netflix algorithm competition [16], the proposal of a matrix factorization algorithm caused a sensation in the industry and had a profound influence on the design of subsequent algorithms and models. In 2010, [PERSON] proposed the FM model [17] using the inner product of two vectors to replace the single weight coefficient, that is, introducing a hidden vector, to better solve the problem of data sparsity. Specifically, let the feature vector be \(x\), weight vector be \(w\), the number of features be \(n\), \(j_{1}\) and \(j_{2}\) be the subscripts of the two features vectors in feature crossing operation respectively and the basic expression of FM be:  
\[\varnothing\text{FM}w,x=\sum\
olimits_{j_{1}=1}^{n}\sum\
olimits_{j_{2}=j_{1} +1}^{n}(w_{j_{1}}w_{j_{2}})x_{j_{1}}x_{j_{2}}. \tag{1}\]  
Using an implicit vector, the number of \(n^{2}\) weight parameters is directly reduced to \(nk\) (where \(k\) is the dimension of the hidden vector), which can significantly reduce the training cost when using the gradient descent. The FM model is a key model in the entire field of recommendation systems. In this study, the basic concept of the FM model was applied to increase the overall efficiency. In 2015, AutoRec [18], proposed by the Australian National University, officially opened an era of deep learning for recommendation systems by combining autoencoders and collaborative filtering. The AutoRec model is similar to MLP (Multi-layer Perceptron), which is a standard 3-layer (including input layer) neural network, but it combines the ideas of AutoEncoder and Collaborative Filtering. To be more precise, the AutoRec model is a standard autoencoder structure, and its basic principle is to use the co-occurrence matrix in collaborative filtering to complete the self-coding of the item vector or user vector. Then the self-coding results are used to obtain the user's score on all items, and the results can be used for item recommendation after sorting. Since then, various models and algorithms have mushroomed, and the deep crossing model [19] proposed by Microsoft in 2016 is one of the sources of inspiration for this study. Its biggest progress is to change the traditional method of feature crossing so that the model is not only capable of second-order crossing, but also of deep crossing. The multiple residual unit layer proposed in this study adopts a multilayer residual network, which enables the model to capture more information on nonlinear and combination features, giving full play to the advantages of multifield crossover technology. Another important article in 2016 was The Wide and Deep Model proposed by Google [20], which proposed the concepts of _memory_ and _generalization_ for the first time, breaking the thinking of traditional models and directly developing a special system. The ARFM proposed in this study are also an evolution of a wide and deep model, to a certain extent. Memory ability can be understood as the ability of a model to directly learn and use the co-occurrence frequency of items or features in the historical data. This can be understood as the ability of the model to transfer the correlation of features and discover the correlation between rare features that are sparse or even never appear and the final tags. The wide part is responsible for the memory of the model, whereas the deep part is responsible for the generalization of the model. This method of combining the two parts of the network structure combines the advantages of both sides, and has become absolutely popular at this time.