## 4 Experiments
### Experimental Results  
The visualization of the 2D embeddings for the synthetic dataset are given in Figure 2. According to this figure, the proposed method provides better class separation as compared with LPP. Furthermore, Table 2 shows the classification results for hyperspectral images. The most successful linear manifold learning method is shown in bold type. The proposed method is implemented for both LPP and OLPP. According to Table 2, KSC has higher general classification accuracy rates than Indian Pines when all manifold learning algorithms used are considered. Proposed + LPP and Proposed + OLPP are the best linear manifold learning algorithms for Indian Pines and KSC, respectively. Furthermore, the most successful linear manifold learning algorithm among the existing ones are OLPP for Indian Pines and NPE for KSC datasets. The enhancement rates are 1.6% and 1.4% for Indian Pines and KSC datasets, respectively. Figure 2: Visualizations of subspaces obtained upon “3d_clusters” toy dataset: (**a**) Original data; (**b**) LPP; (**c**) Proposed + LPP, (\(x\)\(y\) and \(z\) represent the features; LPP: Locality Preserving Projection). Class-based \(\mu_{c}\) values and enhancement rates for every dataset are reported and shown in Figure 3. According to this figure, the number of classes that show decrement in accuracy is only two out of 29 classes. Furthermore, in Figure 3a, a correlation pattern is noted. Another remarkable point is the classes that seem to match with zero \(\mu_{c}\), as shown in Figure 3b. In fact, these are not zero, but very low values as compared with other classes. This occurs because the classes having very similar band information that causes low scatter values. The other part of the experimental results consists of some visualizations for analytical evaluations. First, the weight matrices are extracted. The obtained weight matrices are shown with Figure 4 for hyperspectral datasets. These matrices show the effect of the changes that the proposed method causes. According to these figures, the values of coefficients around diagonals, i.e., the similarities for intra-class values, are increased. Additionally, off-diagonal values for non-intra-class values are decreased. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & **Linear Manifold Learning Method** & **OA (\%) \(\pm\) std** & **Parameters** \\ \hline \multirow{6}{*}{Indian Pines} & LPP & 69.2 \(\pm\) 1.25 & \(k\) = 50, \(s\) = 1600 \\  & NPE & 73.2 \(\pm\) 0.25 & \(k\) = 70 \\  & OLPP & 76.6 \(\pm\) 0.72 & \(k\) = 90, \(s\) = 800 \\  & ONPE & 74.7 \(\pm\) 0.67 & \(k\) = 40 \\  & **Proposed + LPP** & **78.2 \(\pm\) 0.77** & \(k\) = **110**, \(s\) = **400**, \(p\) = **10\({}^{-4}\)** \\  & Proposed + OLPP & 75.2 \(\pm\) 1.25 & \(k\) = 90, \(s\) = 1000, \(p\) = 10\({}^{-4}\) \\ \hline \multirow{6}{*}{KSC} & LPP & 82.9 \(\pm\) 2.05 & \(k\) = 2, \(s\) = 19,500 \\  & NPE & 90.0 \(\pm\) 0.82 & \(k\) = 32 \\ \cline{1-1}  & OLPP & 87.4 \(\pm\) 2.10 & \(k\) = 18, \(s\) = 36,000 \\ \cline{1-1}  & ONPE & 87.0 \(\pm\) 1.00 & \(k\) = 60 \\ \cline{1-1}  & Proposed + LPP & 85.0 \(\pm\) 0.87 & \(k\) = 50, \(s\) = 19,500, \(p\) = 10\({}^{-6}\) \\ \cline{1-1}  & **Proposed + OLPP** & **91.4 \(\pm\) 0.83** & \(k\) = 20, \(s\) = 20,000, \(p\) = 10\({}^{-6}\)** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Classification performance on linear manifold learning methods for the HSI datasets. (OA, overall classification accuracy in percentage; std, standard deviation; \(k\), neighborhood; \(s\), heat kernel parameter; the algorithm with the best performance is bolded for every dataset). Figure 3: Enhancement vs. \(\mu_{c}\) graphs upon best resulting algorithms reported in Table 2: (**a**) Indian P.; (**b**) KSC (enhancement, class-based classification accuracy difference in percent between the proposed method + existing manifold learning algorithm and existing manifold learning algorithm; coeff, \(\mu_{c}\), class-based coefficients proposed with this study). (Note that every circle represents a class.