## 4 Experiments
### Synthetic Experiment  
In the synthetic experiment, we adopt the similar method in [31] to produce the data samples for clustering. Firstly, we randomly select five samples from the pure infrared hyper spectral mineral data and form a matrix \(\mathbf{A}_{i}=[\mathbf{a}_{1},...,\mathbf{a}_{5}]\in\mathbb{R}^{321\times 5}\). Then we produce a random vector \(\mathbf{w}_{i}\in\mathbb{R}^{5}\) by using a uniform distribution and obtain a vector by linear combination \(\mathbf{x}_{i}=\mathbf{A}_{i}\mathbf{w}_{i}\). Let the generation of \(\mathbf{w}_{i}\) be repeated 10 times, we can obtain 10 data samples from the subspace \(span(\mathbf{A}_{i})\), denoted by \(\mathbf{X}_{i}=[\mathbf{x}_{1},...,\mathbf{x}_{10}]\in\mathbb{R}^{321\times 10}\). If we repeat the process of \(\mathbf{A}_{i}\) five times, we finally get a dataset derived from the union of five subspaces, denoted by \(\mathbf{X}=[\mathbf{X}_{1},...,\mathbf{X}_{5}]\in\mathbb{R}^{321\times 50}\), which will be used as the dataset for clustering. To further evaluate the robustness of the proposed methods, we add Gaussian noise onto \(\mathbf{X}\) with three different magnitudes (10%, 20%, 30%) and so we produce three noisy datasets for clustering experiments. To get stable results, the experiments on different datasets are repeated 50 times with the reproduced \(\mathbf{X}\). The minimum, maximum, median and mean clustering errors are recorded for complete evaluation. In the above experiments, the parameters \(\lambda_{1}=20,\lambda_{2}=1\) for LLE-SSC and \(\lambda_{1}=100,\lambda_{2}=1000\) for LLE-LRR.  
The experimental results are shown in Table 2. In the noise free case, almost all clustering methods perform well; SSC, SMR, LapLRR and our model LLE-LRR have completely correct results. However, when the noise amplitude increases, the clustering performances of the compared methods decrease rapidly, while that of our methods drops slowly. It is shown that compared with the other methods, our LLE-SSC method has at least 10% improvements in metric of median and at least 8% improvements in metric of mean when the noise level is 30%, which highlights that our methods have good robustness to noise.  
For the subspace clustering methods, the constructed affinity matrix is critical for clustering. The ideal affinity matrix will be a block diagonal form. Generally, the closer to the block diagonal form it is, the better the clustering results will be. Therefore, we provide a visual comparison of affinity matrices produced by different methods in case of adding 20% Gaussian noise, as shown in Figure 1. It is observed that our proposed methods, LLE-SSC and LLE-LRR, have affinity matrices closer to block diagonal shape than other methods. Additionally, we further show the clustering results by visualizing the samples of the same class with the same color, as shown in Figure 2. It is indicted that our methods, LLE-SSC, LLE-LRR and NSLLRR, have the best clustering results.  
\begin{table}
\begin{tabular}{l l c c c c c c c} \hline \hline \multirow{2}{*}{**Gaussian Noise\%**} & \multicolumn{6}{c}{**The Clustering Error**} \\ \cline{2-10}  & \multirow{2}{*}{**SSC**} & **LRR** & **NSLLRR** & **SMR** & **LapLRR** & **LLE-SSC** & **LLE-LRR** \\ \hline \multirow{4