## 2 Methods
### Graph Embedding  
GE [30] is used as a distribution difference constraint to measure the difference between domains by maintaining a similar relationship between samples in the neighborhood. \(x_{i}\) can be linearly represented by its neighborhood samples \(x_{j}\), \(x_{k}\), and \(x_{l}\), which can be shown as follows  
\[x_{i}=w_{ij}x_{j}+w_{ik}x_{k}+w_{il}x_{1} \tag{9}\]  
We give greater weight to the points with high similarity \(w_{ij}\). To retain the structure information of the data, the graph embedding method treats each sample vector as a vertex. By calculating the distance between the points, the neighboring points are assigned higher weights, and the non-adjacent points are assigned lower weights. Finally, a similarity matrix \(W\) describes the geometric characteristics of the data. In order to keep the local structural consistencies on labels, true and pseudo labels are used to structure the graph. We introduce inter-database similarity matrices \(W_{st}\) and \(W_{st}\). It is assumed that if \(x_{i}^{s}\) and \(x_{j}^{t}\) pertain to the same class, and \(x_{i}^{s}\)\(x_{j}^{t}\) are the k nearest neighbors of \(x_{j}^  
\[W=\left[ \begin{array}{cc}W^{ss}&W^{st}\\ W^{ts}&W^{tt}\end{array} \right] \tag{14}\]  
Each element of \(W\) represents the similarity between the samples. Finally, the objective function of graph embedding is expressed as  
\[ \begin{split} G(K)&=\frac{1}{2}\sum_{i,j=1}^{N}\|x_{i}-x_{j}\|^ {2}w_{ij}\\ &=\frac{1}{2}\left(\sum_{i=1}^{N}x_{i}^{2}\sum_{j=1}^{N}w_{ij}+\sum_{j=1}^ {N}x_{i}^{2}\sum_{i=1}^{N}w_{ij}-2\sum_{i=1}^{N}\sum_{j=1}^{N}x_{i}x_{j}w_{ij} \right)\\ &=\sum_{i=1}^{N}x_{i}^{2}D_{ii}-\sum_{i=1}^{N}\sum_{j=1}^{N}x_{i}x_{j}w_{ ij}\\ &=\text{tr}\Big{(}P^{T}XLX^{T}P\Big{)}\end{split}  \tag{15}\]  
where \(L\) is a Laplacian matrix, which can be expressed as \(L=D-W\), and \(D\) is a diagonal matrix whose entries are the column sums of \(W\). In combination with the other part, the final optimization function can be written as  
\[\text{min}\sum_{c=0}^{c}\text{tr}\Big{(}A^{T}\Big{(}(XM_{c}+\mu L)X^{T}+ \lambda(S_{w}-S_{b})+I\Big{)}A\Big{)}+\|A\|_{F}^{2} \tag{16}\]  
Then, the Lagrange multiplier method is employed to further optimize this formula  
\[L=\text{tr}\Bigg{(}A^{T}\Bigg{(}X\Bigg{(}\sum_{c=0}^{C}W_{c}+\mu L\Bigg{)}X^{ T}+\lambda(S_{w}-S_{b})+I\Bigg{)}A\Bigg{)}+\text{tr}\Big{(}\Big{(}I-A^{T}XHX^{T}A \Big{)}\phi\Big{)} \tag{17}\]  
Considering \(\frac{\text{d}L}{\text{d}A}=0\), the generalized eigen decomposition is achieved as follows:  
\[\Bigg{(}X\Bigg{(}\sum_{c=0}^{C}W_{c}+\mu L\Bigg{)}X^{T}+\lambda(S_{w}-S_{b})+ I\Bigg{)}A=XHX^{T}A\phi \tag{18}\]  
By solving this equation, we obtain a mapping matrix \(A\), which acts on the source and target domains. Then, two close low-dimensional feature spaces are obtained for the source and target database, respectively. Finally, a traditional machine learning classifier is trained on the labeled source database and applied to an unlabeled target database for pathological voice detection and classification. Algorithm 1 presents the complete flow of the JSTL. ```
1:Input: source and target data \(X\); source domain labels \(y_{s}\); regularization parameter \(\lambda\), \(\mu\);
2:Output: target domain labels \(y_{t}\)
3:\(M_{0}\), \(S_{w}\), \(S_{b}\)
4:repeat until convergence
5: construct the graph matrix by true and pseudo label
6: solve eigendecomposition \(X\Big{(}\sum_{c=0}^{C}W_{c}+\mu L\Big{)}X^{T}+\lambda(S_{w}-S_{b})+I\Big{)} A=XHX^{T}A\phi\) and \(k\) smallest eigenvectors as adaptation matrix \(A\)
7: update pseudo target labels using a standard classifier \(f\) trained on projected source data
8: update \((M_{c})_{ij}\) and \(L\)
9: end repeat
10: return target domain labels \(y_{t}\) determined by classifier

## 3 Experimental Methodology

In this section, we introduce three databases commonly used in pathological voice research [31], feature set, and feature selection. ### Database

The cross-database training samples were selected from three pathological voice database.