# Collaborative Filtering Based on a Variational Gaussian Mixture Model
###### Abstract  
Collaborative filtering (CF) is a widely used method in recommendation systems. Linear models are still the mainstream of collaborative filtering research methods, but non-linear probabilistic models are beyond the limit of linear model capacity. For example, variational autoencoders (VAEs) have been extensively used in CF, and have achieved excellent results. Aiming at the problem of the prior distribution for the latent codes of VAEs in traditional CF is too simple, which makes the implicit variable representations of users and items too poor. This paper proposes a variational autoencoder that uses a Gaussian mixture model for latent factors distribution for CF, GVAE-CF. On this basis, an optimization function suitable for GVAE-CF is proposed. In our experimental evaluation, we show that the recommendation performance of GVAE-CF outperforms the previously proposed VAE-based models on several popular benchmark datasets in terms of recall and normalized discounted cumulative gain (NDCG), thus proving the effectiveness of the algorithm.  
recommendation system; variational autoencoder; collaborative filtering; Gaussian mixture model +
Footnote †: journal:  
0000-0001-9991-4000-0000-0000-0000-0000-0000-0000-0000-0000-0000-00000-0000-00000-00000-00000-00000-00000-00000-0000-00000-  
for the side features vector of probabilistic matrix factorization (PMF), and this Bayesian constrained PMF model is superior to simple MAP estimation.  
However, MF cannot capture the non-linearity relationships between users and items. To tackle this problem, the generative models have attracted the attention of researchers. The first generative model that appeared in the literature of recommendation systems was the generative adversarial network (GAN) [10; 11]. Immediately afterwards, a series of variational autoencoders (VAEs) also appeared, and they achieved better results than using GAN for recommendation systems [12; 13]. [PERSON] et al. [14] used VAEs for the first time to conduct CF by extending VAEs to CF recommendation algorithms for implicit feedback, which made up for the shortcomings of less research on non-linear models in recommendation systems. [PERSON] et al. [15] modified the distribution of the latent factors in the original VAE and used a flexible prior distribution, and said that this method can better represent the user preferences. [PERSON] et al. [16] introduced a recommender VAE (RecVAE) model that uses a novel composite prior distribution for the latent representation. The distribution of the hidden variables of the VAE model in the above CF algorithm is too simple, and it cannot effectively learn the complex non-linear relationships between users and items. In summary, our contributions are as follows:  
* We propose a new variational autoencoder CF model (i.e., a Gaussian mixture model for latent factor distribution for CF (GVAE-CF)). Specifically, we change the single Gaussian posterior distribution in the standard variational autoencoder to a Gaussian mixture model (GMM), and we propose a GMM-based variational autoencoder neural network structure for CF. The model can effectively learn the relationships of users and items.
* We re-derive the variational lower bound of the variational autoencoder.
* We compare the latent space representations of the standard VAE and GVAE-CF to study the GVAE-CF internal mechanism of the GVAE-CF.