## 5 Conclusions  
In this paper, we proposed a new data clustering framework based on manifold learning and subspace clustering. Different from the traditional data subspace methods, which try to utilize the data similarity to construct Laplacian graph constraint for data representation. In the proposed framework, LLE is used to learn the manifold structure hidden in the data. The learned non-linear manifold is taken as a regularization for the subspace clustering methods, SSC and LRR, which results in the proposed LLE-SSC and LLE-LRR clustering methods. To evaluate the proposed clustering methods, we implement several clustering experiments on different types of datasets, such as the synthetic data, face images, handwritten digital images, object images and the motion sequences. The experimental results showed that the proposed methods have good clustering performance compared with related methods on different datasets. Particularly, the proposed methods have two advantages. Firstly, from the results of the experiments on the synthetic data with noise, it is shown that the proposed methods are robust \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Number of Classes**} & \multicolumn{6}{c}{**The Clustering Error (\%)**} \\ \cline{2-7}  & **SSC** & **LRR** & **NSLLRR** & **SMR** & **LapLRR** & **LLE-SSC** & **LLE-LRR** \\ \hline
2 & 5.32 & 6.38 & 2.92 & **2.13** & 2.50 & 3.10 & 3.33 \\
3 & 10.59 & 7.75 & 9.48 & 6.98 & 8.55 & **5.12** & 6.42 \\
4 & 14.05 & 12.55 & 10.12 & 16.89 & 11.48 & **7.25** & 8.87 \\
5 & 21.70 & 15.29 & 15.20 & 17.18 & 14.61 & **10.37** & 10.89 \\
6 & 25.31 & 17.84 & 16.20 & 16.03 & 16.78 & 16.13 & **12.38** \\
7 & 29.07 & 21.31 & 20.61 & 18.54 & 18.70 & 16.79 & **14.06** \\
8 & 25.63 & 22.21 & 24.91 & 21.64 & 16.75 & 16.42 & **14.53** \\
9 & 27.54 & 25.32 & 24.87 & 23.69 & **19.22** & 23.90 & 20.93 \\
10 & 33.21 & 27.52 & 25.40 & 24.09 & 21.46 & 18.93 & **17.60** \\
11 & 30.21 & 29.49 & 28.96 & 25.27 & 25.38 & 26.31 & **20.72** \\ mean & 22.26 & 18.57 & 17.87 & 17.24 & 15.54 & 14.43 & **12.97** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Clustering results on COIL20 dataset. The clustering errors for \(2,3,4,5,6,7,8,9,10,11\) classes and their mean error are shown here. The best results are in bold font. Figure 6: Some object images from the COIL20 dataset. to the noisy data. Secondly, from the clustering results on the Extended Yale B, USPS, COIL20 datasets with the number of the clusters increasing, we can observe that our methods show more stability than the compared methods, which means our methods have the potential for clustering a large scale of datasets. These characteristics make them prospective methods in data clustering. It is shown that using the manifold learning method to reveal the non-linear property of the data and introducing the learned manifold constraint into subspace clustering methods will improve the data clustering performance. However, in this paper, we only use the common used LLE method to verify this idea by integrating it with the classic subspace clustering methods, SSC and LRR. We believe that the developing manifold learning methods are more helpful in dealing with more complicated data in practice. So one of our future works is to exploit a more suitable manifold learning method and integrate it to improve the data clustering performance. The other future work is to extend our methods by parallel computation technique in order to improve their efficiency and make them practical and suitable for clustering large scale data. Conceptualization, methodology, writing review and editing, contributed by [PERSON]; Investigation, validation, formal analysis, contributed by [PERSON]; Software, data curation, visualization, writing original draft preparation, contributed by [PERSON]. This research was funded by National Natural Science Foundation of China under Grant Nos. 61390510, 61672071, 61632006, 61772048, Beijing Natural Science Foundation Nos.