Locality-preserving projection (LPP) has been widely used in feature extraction. However, LPP does not use data category information and uses the ${L}_{2}$ -norm for distance measurement, which is highly sensitive to outliers. In this paper, we consider the LPP weight matrix from a supervised perspective and combine the low-rank regression method to propose a new model to discover and extract features. By using the ${L}_{2,1}$ -norm to constrain the loss function and the regression matrix, not only is the sensitivity to outliers reduced but the low-rank condition of the regression matrix is also restricted. Then, we propose a solution to the optimization problem. Finally, we apply the method to a series of face databases, handwriting digital datasets and palmprint datasets to test the performance, and the experimental results show that this method is effective compared with some existing methods.

Traditional multi-view algorithms typically require data to be complete, and these algorithms may not be suitable or effective when dealing with incomplete data. As a result, the research field has witnessed the emergence of methods specifically designed for addressing incomplete multi-view clustering. In contrast, most of existing incomplete multi-view algorithms primarily emphasize capturing global information while neglecting the importance of local information. To overcome these challenges, we put forward a novel approach, named tensor low-rank graph embedding and learning for one-step incomplete multi-view clustering. We combine the beneficial aspects of graph embedding and tensor low-rank in this method, which not only focuses on local relationships but also provides insights into the global structure. Firstly, the initial similarity graph matrix is constructed using the inter-dependency between views. Secondly, the similarity graph matrix is structured as a third-order tensor and constrained by the tensor nuclear norm. This constraint enables the capturing of higher-order correlations across multiple views. Thirdly, graph embedding is added to obtain a common feature representation. Finally, the algorithm incorporates clustering to determine the optimal clustering labels. We conducted experiments comparing our algorithm with seven different incomplete multi-view methods using four different evaluation metrics. The experimental results indicate that our algorithm achieved the best clustering performance across five datasets under various missing rates. Especially when faced with a 50% missing rate in the Extended YaleB dataset, the clustering accuracy and normalized mutual information of our method are improved by 28.23% and 28.11%, respectively, compared to the second-best algorithm.