## 1 Introduction  
In the era of information explosion, big data exhibits a rich value and great potential, which brings transformative development to human society, but it also generates the serious \"information overload\" problem. Recommender systems are an effective way to alleviate the problem of \"information overload\", having been widely adopted by many online services, including E-commerce, online news, and social media sites [1]. Recommender systems can help determine which information to offer to individual consumers and allow online users to quickly find the personalized information that fits their needs [2]. Collaborative Filtering (CF) is a successful approach commonly used by many recommender systems [3]. CF [4; 5] is based on the user's past interaction records (such as ratings) to simulate the user's preferences for the item. The scarcity of original interaction records has always been a difficult point for CF, which is the problem of data sparsity. As the most popular approach among various CF techniques, Matrix Factorization (MF) [6; 7], which projects users and items into a shared latent space, using a vector of latent features to represent a user or an item, has become a standard model for recommendation due to its scalability, simplicity, and flexibility [8; 9]. Most previous research [3; 7; 10; 11; 12; 13; 14; 15] on MF did not change the nature of linearity, such as integrating it with neighbor-based models [7], combining it with topic models of item content [3], and extending it to factorization machines [15] for a generic modelling of features. Despite the effectiveness of MF for CF, such a linear model is insufficient to understand the complex and non-linear relationship between users and items [16]. Furthermore, the problem of data sparsity is difficult to solve in MF. Of course, there has been some advanced research [1; 2; 17; 18] on MF that essentially changed the linear structure of MF by combining the idea of deep learning. The combination of deep learning and MF mainly uses the implicit feedback data to learn the hidden vector of the user(item) by using the deep learning method, thereby predicting the user's preference for the item based on the hidden vector. Among existing models, an outstanding performance was shown for the NeuMF (Neural Matrix Factorization) model proposed in [1]. The NeuMF model combines matrix factorization and a multi-layer perceptron to achieve a good performance. MF based on deep learning can be regarded as a kind of nonlinear generalization of the traditional hidden factor model. Its obvious advantage is the introduction of nonlinear feature transformations in the process of learning the hidden vector of the user(item), which has a better performance than traditional MF. However, this method still cannot improve the problem of data sparsity in MF. In this paper, in order to alleviate the problem of data sparsity, we propose a new CIDAE (Continuous Imputation Denoising Autoencoder) model based on the Denoising Autoencoder (DAE). The CIDAE model performs regular continuous imputation for the missing parts of the original data and trains the imputed data as the desired output, which is different from the DAE algorithm that randomly adds noise to the input data. The advantage of CIDAE is that it alleviates the problem of data sparsity by using the idea of imputation. Then, this paper optimizes the advanced NeuMF model based on the training process. The NeuMF model introduces the idea of deep neural networks into matrix factorization. The NeuMF model first proposes the GMF (Generalized Matrix Factorization) model and then combines the GMF model with the MLP (Multi-Layer Perceptron) model. We name the optimized NeuMF model the O-NeuMF-p model. Additionally, our experimental results prove that the accuracy of O-NeuMF-p is higher than that of NeuMF. Finally, we fuse CIDAE and O-NeuMF-p with reference to the idea of ensemble learning. We name the fused model the I-NMF (Imputation-Neural Matrix Factorization) model, which combines the advantages of CIDAE and O-NeuMF-p.