## 4 Experiments
### Experimental Settings  
The experiments are conducted on hyperspectral images; but, first, the effect of the proposed method is shown on synthetic data. LPP and Proposed + LPP embeddings are used, and the parameters are chosen heuristically to reduce three dimensions to two dimensions. Then, the classification accuracy performances that belong to LPP, OLPP, NPE, ONPE, the proposed method + LPP, and the proposed method + OLPP are searched on hyperspectral images. The reduced dimensionality (RD) is fixed to 20. The optimum neighborhood \(k\) and heat kernel parameter \(s\) are searched based on the classification performance of each algorithm that achieves its maximum over a range. The boundaries of the search loops and interval values for every iteration are determined heuristically. All experiments are repeated five times with random samples and classification rates are averaged. The overall accuracy values are shown with tables in the next subsection. The algorithm flowchart for the classification experiments on a randomly selected train and test data is illustrated in Figure 1.  
As shown in Figure 1, the first step is to split the data into training and test datasets. The training dataset is employed for dimensionality reduction learning. A transformation matrix is created and with the help of this matrix, training and test embeddings are constructed. The process shown in the red rectangle continues over a range of \(s\) and \(k\). The accuracy values are recorded during this period. Training and test dataset sample numbers are given with Table 1. The classes belonging to the HSI datasets do not have the same number of samples.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & **Indian P.** & **KSC** \\ \hline Total & 10,249 & 5211 \\ \hline Training & 1031 & 528 \\ \hline Test & 9218 & 4683 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Train/test sample numbers of the HSI datasets used.  
Figure 1: The flowchart used for classification. (Red rectangle contains the process on a selected training/test dataset for feature learning. This part is repeated five times with randomly selected training/test datasets and classification results are kept, and then averaged to find optimal \(k\) and \(s\). Cross validation is applied to train embeddings to determine hyperparameters. Then, the test data is evaluated. The test data are completely unseen data and used just for prediction).  
Support vector machine (SVM) is chosen as the classifier in this study. In the SVM, a radial basis function (RBF) kernel is used with the penalty parameter, \(C\in[2^{-4},\,2^{8}]\) and kernel width parameter, \(\gamma\in[2^{-4},\,2^{4}]\). A grid search is applied on the closed intervals indicated. Cross validation is applied to search parameters.  
Another main part of the study is to conduct experiments for analytical evaluations. The manifold learning algorithm that yields the highest classification accuracy performance is taken into consideration for the analysis to show the effects of the proposed method, which is the experimental strategy. The best resulting algorithm may become LPP or OLPP with the proposed versions, depending on the dataset.  
The class-based accuracy enhancements and coefficients, \(\mu_{c}\), that are proposed with this study are visually reported. The other analytical elements can be expressed as the evaluation of band information, weight, and correlation matrices with their significance. The analytical experiments are carried out with randomly chosen data by considering the sample rates shown in Table 1. The parameters remained the same as previously determined in classification phase.  
Classes 3 and 12 for Indian Pines and Classes 2, 7, 8, and 9 for KSC are declared as difficult to discriminate [35]. In the band information experiments, these classes are evaluated based on the first two bands. Furthermore, Bands 11 and 12 are chosen empirically to search whether they provide discrimination for any class.