## 5 Conclusions  
This paper presents a novel approach for ranking support vector machines, which is based on manifold regularization matrix factorization. The proposed method combines  
Figure 6: The performance of our proposed method on the Scene dataset through pair-wise comparison for parameters \(\alpha\), \(\gamma\) and \(\theta\). (**a**) Example-F1 for (\(\gamma\), \(\theta\)) pair, (**b**) Example-F1 for (\(\alpha\), \(\theta\)) pair, (**c**) Example-F1 for (\(\alpha\), \(\gamma\)) pair, (**d**) Micro-F1 for (\(\gamma\), \(\theta\)) pair, (**e**) Micro-F1 for (\(\alpha\), \(\theta\)) pair, (**f**) Micro-F1 for (\(\alpha\), \(\gamma\)) pair, (**g**) Hamming Los for (\(\gamma\), \(\theta\)) pair, (**h**) Hamming Los for (\(\alpha\), \(\theta\)) pair, and (**i**) Hamming Los for (\(\alpha\), \(\gamma\)) pair. Figure 7: The evolution of the objective values of our proposed method on the Scene (**a**) and Yeast (**b**) datasets. feature selection with multi-label learning, creating a unified framework. Within this framework, matrix factorization is introduced to learn low-rank robust subspaces in the input space, thereby enhancing the robustness of data representation in the presence of high levels of noise. Furthermore, by preserving the manifold relationship among low-rank samples, the framework incorporates manifold structure regularization to further improve the robustness of the low-rank representation. Subsequently, using this robust low-rank representation, we extract low-order features that are more resilient and employ them to construct superior classifiers. In other words, these methods enable our classification model to perform better in the presence of original noise. To validate the feasibility of the proposed algorithm, we conducted extensive experiments on real-world multi-label datasets. We compared our algorithm against five commonly used multilabel classification algorithms using multiple evaluation metrics. The experimental results demonstrate that the proposed method outperforms state-of-the-art methods across all evaluation metrics. In the future, we plan to introduce deep learning methods to further enhance multi-label classification models. To further develop our approach, firstly, different models may be appropriate for different fields. Regardless of the data type, the goal of feature extraction is to find the most representative and informative features in order to improve the performance of machine learning models. The dimensions, processing techniques, and final feature composition that need to be considered vary for different types of data. Therefore, the adaptation of different models on different datasets is a question worthy of further research. Secondly, when dealing with the increasingly large and complex data, the generalization ability and scalability of our model may need to be improved. [PERSON] et al. [32] introduced graph convolutional networks to handle multi label image recognition tasks, utilizing the relationships between labels to improve classification performance. [PERSON] et al. [33] proposed a deep convolutional network to handle this incomplete multi label learning problem. These methods are very inspiring for us, and in the future, we can also introduce deep learning methods to further improve multi label classification models. Conceptualization, H.S., Y.Z. and X.S. ; Methodology, H.S., Y.Z. and X.S. ; Software, Y.Z., E.Q. and Q.Z. ; Supervision, H.S. and X.S. ; Writing--original draft, H.S., Y.Z., E.Q., Q.Z. and X.S. ; Writing--review and editing, H.S., Y.Z. and X.S. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Natural Science Foundation of China under grant 62172193 and the Yangzhou Industrial Foresight and Common Key Technology Projects under grant SCY2023000087. Not applicable. Publicly available datasets were analyzed in this study.