## 5 Conclusions  
This paper introduces a recommendation algorithm based on an improved Conditional Variational Autoencoder (CVAE) and Constrained Probabilistic Matrix Factorization (CPMF). In this algorithm, a hidden layer is added on top of the CVAE architecture, and random noise is introduced into the hidden layer to enhance the model's robustness against interference. The processing of item data through the hidden layer further enhances the model's generative capability. Additionally, CVAE incorporates item categories as auxiliary information to supervise the encoding and decoding of project data. By learning the distribution features of the data, the algorithm reconstructs missing values in the rating data, effectively mitigating data sparsity.  
Furthermore, the reconstructed data is subjected to the CPMF algorithm to optimize feature extraction by mining the latent implicit rating features of users and items. Finally, the predicted ratings are obtained through the multiplication of user and item feature matrices. Experimental validation is conducted on the MovieLens-100K and MovieLens-1M datasets, and a comparison is made against four alternative algorithms. The experimental results demonstrate the superior recommendation performance of the proposed algorithm. This indicates that the proposed algorithm effectively addresses the issues of poor recommendation performance often encountered in traditional user-based collaborative filtering algorithms due to data sparsity and suboptimal feature extraction.  
However, it is worth noting that the current algorithm utilizes a singular auxiliary information source to guide item outputs. In future research, the incorporation of multiple auxiliary information sources, such as knowledge graphs, attention mechanisms, and social networks, could be explored to enhance user and item features and further elevate recommendation performance.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Comparison Method** & **RMSE** & **MAE** \\ \hline CDL & 0.9234 & 0.7485 \\ CTR & 0.9338 & 0.7588 \\ PHD & 0.9265 & 0.7475 \\ PMF & 0.9520 & 0.7755 \\ This Paper & 0.9008 & 0.7222 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of Different Methods in MovieLens-100K.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Comparison Method** & **RMSE** & **MAE** \\ \hline CDL & 0.8922 & 0.7183 \\ CTR & 0.9032 & 0.7272 \\ PHD & 0.8905 & 0.7163 \\ PMF & 0.9082 & 0.7452 \\ This Paper & 0.8483 & 0.6855 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Results of Different Methods in MovieLens-1M.  
**Author Contributions:** Writing--original draft, Y.Z.; Writing--review & editing, H.X. and X.Y. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work is supported by the Fuzhou Talent Program of Jiangxi Province (2021 ED008), and Jiangxi Key Laboratory of Cyberspace Security Intelligent Perception (JKLCIP202202).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Publicly available datasets were analyzed in this study. This data can be found here: [[[https://grouplens.org/datasets/movielens/](https://grouplens.org/datasets/movielens/)]([https://grouplens.org/datasets/movielens/](https://grouplens.org/datasets/movielens/))].  
**Conflicts of Interest:** The authors declare no conflict of interest.