## 3 The Proposed Method  
The LE attempts to preserve the local structure of data. First, it constructs an adjacency graph, and then finds the weight matrix. The heat kernel function defined in Equation (3) can be applied to the graph at this phase. These steps are common for both the LPP and the OLPP. A supervised version of the LE is overviewed in a study by [32]. It finds within-class and between-class neighbors upon a specific similarity threshold value. The algorithm continues by constructing two different weight matrices for within and between classes, \(W_{w}\) and \(W_{b}\). At these steps, the heat kernel function is constructed with a preselected constant heat kernel parameter, \(t\). The next step is the optimization of two objective functions, i.e., maximizing \(\sum_{i,j}(y_{i}-y_{j})^{2}W_{b,\ ij}\) and minimizing \(\sum_{i,j}\bigl{(}y_{i}-y_{j})^{2}W_{w,\ ij}\). Finally, these are combined to give a generalized eigenvalue solution. In the proposed method, the weight matrix, \(W\), is constructed with the help of a changing parameter in the denominator of Equation (3). The denominator is formulated based on class-specific scattering information of training data with a supervised manner. The proposed method optimizes an objective function depending on this \(W\). It means that the proposed method requires only one weight matrix instead of two for optimization. The method is applied on the LPP and the OLPP for classification purpose. Let \(\mu_{c}\) be the coefficient that keep the specific values calculated for every class with \(c\) in 1, 2,..., \(n\) and \(n\) is the total class number, and \(D\) is the \(m\times m\) distance matrix where \(m\) is the number of training samples. Then, \(\mu_{c}\) is found by applying Equations (8) and (9), sequentially, based on the information of \(D\) as follows:  
\[S_{i}=\sqrt{\frac{1}{k_{i}-1}\sum_{j=1}^{j=k_{i}}\bigl{(}D\bigl{(}i,I_{ij} \bigr{)}-avg_{i}\bigr{)}^{2}} \tag{8}\]  
\[u_{c}=\frac{1}{t_{c}}\sum_{j=1}^{j=t_{c}}\left(s_{i_{group,c}}\right)_{j} \tag{9}\]  
Equation (8) calculates the standard deviation of every \(i\)-th sample of \(D\) considering only intra-class distance information. Here, the \(i\)-th sample means the \(i\)-th row and \(i\) changes from \(1\) to \(m\). The output of the equation is shown by \(S_{i}\) which corresponds to the \(i\)-th row. In the equation, \(k_{i}\) keeps the total number of intra-class samples for every row. \(I_{ij}\) keeps the intra-class sample indices for every \(i\)-th row; the maximum number of \(j\) dependson the \(i\)-th sample intra-class number, \(k_{i}\). The average of distances of intra-class samples of \(i\)-th rows is shown with \(avg_{i}\). After \(S_{i}\) is calculated for every row of \(D\), the values that belong to the same classes on \(S\) are grouped by holding the indices of \(S\) elements. This information is kept on \(s_{I_{grouped,c}}\) and allowed us to calculate the average of each grouped class as given by Equation (9). The indices of \(S\) elements belonging to the same classes are represented by \(I_{grouped,c}\), where \(c\) is for class id. The total sample number of every class is shown by \(t_{c}\). The maximum number of \(j\) depends on the sample number belonging to the specific classes, \(t_{c}\). At the end, \(n\) different \(\mu\) are obtained that are represented as \(\mu_{c}\). Due to the symmetrical structure of \(D\), the explanations made for notations are also valid for the column-wise perspective. Up to this point, the procedure to find \(\mu_{c}\) was explained. This parameter is exclusive for each class of training samples and used in the formulation to find \(W\). The main steps of the application of the proposed method to LPP and OLPP are described as follows:  
* The adjacency graph is constructed. * Equation (10) is applied to samples among k-nearest neighbors. \[W_{ij}=\left\{ \begin{array}{c}e^{-\frac{||x_{i}-x_{j}||^{2}}{t}\frac{1}{n_{ c}^{2}p}}\;,\;if\;c(x_{i})=c(x_{j})\\ e^{-\frac{||x_{i}-x_{j}||^{2}}{t}}\;,\;otherwise\end{array}