## 4 Experiments  
To evaluate the performance of the proposed methods, we implement clustering experiments on several public databases. Firstly, to test the proposed clustering methods in an ideal situation, we use a dataset with true subspace structure for clustering. Here, the synthetic dataset in [31] is adopted,which can be regarded as the baseline dataset for the subspace clustering methods. Then we test the proposed clustering methods on several real world datasets, including the Extended Yale B face dataset [32], the handwritten digit dataset of United States Postal Service (USPS) [33], the image dataset of COIL20 [34] and the motion track dataset, Hopkins155 [35], which are considered challenging for clustering as there is too much variety in these datasets.  
The proposed methods, LLE-SSC and LLE-LRR, are compared with several related subspace methods, including the classic SSC and LRR methods, which are considered the baseline methods in subspace clustering methods. Additionally, the proposed methods are also compared with several variations of SSC and LRR, including SMR [14], LapLRR [15] and NSLLRR [17], which introduced other constraints for the data representation, like Laplacian graph, to improve the clustering performance. We summarize the metrics of the data representation error, the constraints for the data representation and the methods for constructing the Laplacian matrix in the related methods and our methods in Table 1.  
To evaluate the performance of these clustering methods, the Subspace Clustering Error (SCE) defined in [31] is adopted as the measurement with the following form:  
\[SCE=\frac{\text{num.misclassified points}}{\text{total num.of points}}, \tag{26}\]  
where \"num.misclassified points\" represents the number of misclassified samples, and \"total num.of points\" is the number of total samples.  
There are two parameters \(\lambda_{1},\lambda_{2}\) which should be set in the proposed LLE-SSC and LLE-LRR methods. However, for the balancing parameters in optimization models, few effective methods are available to obtain the optimal value, as it is generally dependent on the specific datasets. So to get the proper values of these parameters, we adopt a tuning method based on a set of pre-experiments. Concretely, we change the values of these parameters in an estimated range, and for each selected value, we implement some clustering experiments. By comparing the clustering results with different values of these parameters, we can get the favorable values, though they may be not the global optimal values. In the following experiments, we will report the parameters setting for each dataset. The parameters of the compared methods are set as the same as the recommended setting in the original paper or tuned manually based on some pre-experiments similar to the above setting of ours.  
\begin{table}
\begin{tabular}{c} \(\mathbf{L}^{\mathbf{M}}\) is learned by manifold learning \\ method of LLE. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 1: The metrics of the data representation error, the constraints for the data representation and the methods for constructing the Laplacian matrix in the related methods and our methods.