## 2 Methods
### Maximum Mean Discrepancy  
Firstly, the MMD was employed as the discrepancy metric. The aim of the MMD [25] was to map the samples in the source and target domains into a Reproducing Kernel Hilbert Space (RKHS) [28]. Then, we calculated the mean difference between the source and target domains in the RKHS to measure the similarity of different data sets. The smaller the gap, the more similar the distribution; the larger the gap, the greater the difference in distribution. The MMD can be written as follows  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Description** \\ \hline \(X^{s}\in R^{m\times n_{t}}\), \(X^{t}\in R^{m\times n_{t}}\) & Labeled source/unlabeled target features \\ \(M_{0}\in R^{n\times n}\), \(M_{c}\in R^{n\times n}\) & MMD matrix \\ \(L\in R^{n\times n}\) & Laplacian matrix \\ \(D_{s},D_{t}\) & Source/target domain \\ \(S_{w},S_{b}\) & Intraclass/interclass divergence matrix \\ \(c\) & the \(c\)-th pathological voice category \\ \(\lambda\) & regularization parameter of \(S_{w},S_{b}\) \\ \(\mu\) & regularization parameter of GE \\ \(k\) & number of nearest neighbors \\ \hline \hline \end{tabular}
\end{table}
Table 1: Description of frequently used parameters.  
Figure 1: Flowchart of the proposed method.  
\[D(x_{s},x_{t}) =\left\|\frac{1}{n_{s}}\sum_{i=1}^{n_{s}}P^{T}x_{i}-\frac{1}{n_{t}} \sum_{j=n_{s}+1}^{n_{s}+n_{t}}P^{T}x_{j}\right\| \tag{1}\] \[=P^{T}\bigg{(}\frac{1}{n_{s}^{2}}X_{s}I_{s}I_{s}^{T}X_{s}^{T}- \frac{1}{n_{s}n_{t}}X_{s}I_{s}I_{t}^{T}X_{t}^{T}\] \[+\frac{1}{n_{t}n_{s}}X_{t}I_{t}I_{s}^{T}X_{s}^{T}-\frac{1}{n_{t}^{ 2}}X_{t}I_{t}I_{t}^{T}X_{t}^{T}\bigg{)}P\] \[=\text{tr}\Big{(}P^{T}XM_{0}X^{T}P\Big{)}\]  
where \(M_{0}\) is the MMD matrix, which can be written as  
\[\left(M_{0}\right)_{ij}=\left\{\begin{array}{cc}\frac{1}{n_{s}^{2}}&x_{i},x _{j}\in D_{s}\\ \frac{1}{n_{t}^{2}}&x_{i},x_{j}\in D_{t}\\ -\frac{1}{n_{t}n_{t}}&\text{otherwise}\end{array}\right. \tag{2}\]  
Although the MMD can eliminate the divergence in marginal distribution across domains, the difference of the conditional distribution cannot be reduced well. In real research, conditional distribution matters. Therefore, similar to the JDA [29] method, when computing conditional distribution adaptation, the label of the target data is unknown. The solution is to train a set of classifiers on the source data, then apply the classifier to the target data to obtain the labels of the target data. Then, the true and pseudo labels of the data are employed to calculate the conditional distribution. Here, the MMD is modified to measure the class-conditional distributions:  
\[D(x_{s},x_{t}) =\left\|\frac{1}{n_{s}^{c}}\sum_{x_{i}\in X_{s}^{T}}P^{T}x_{i}- \frac{1}{n_{t}^{c}}\sum_{x_{j}\in X_{s}^{c}}P^{T}x_{j}\right\| \tag{3}\] \[=\text{tr}\Big{(}P^{T}XM_{c}X^{T}P\Big{)}\]  
where \(n_{s}^{c}\) and \(n_{t}^{c}\) represent the number of samples in the source and target data which pertain to c class,respectively. \(X_{s}^{c}\) and \(X_{t}^{c}\) are the feature set which pertain to c class in the source and target domains, respectively. \(M_{c}\) is the MMD matrix, which can be written as  
\[\left(M_{c}\right)_{ij}=\left\{\begin{array}{cc}\frac{1}{n_{t}^{2}},&\mathbf{ x}_{i},\mathbf{x}_{j}\in\mathcal{D}_{s}^{(c)}\\ \frac{1}{n_{t}^{2}},&\mathbf{x}_{i},\mathbf{x}_{j}\in\mathcal{D}_{t}^{(c)}\\ -\frac{1}{n_{t}n_{c}},&\left\{\begin{array}{c}\mathbf{x}_{i}\in\mathcal{D}_{ s}^{(c)},\mathbf{x}_{j}\in\mathcal{D}_{t}^{(c)}\\ \mathbf{x}_{i}\in\mathcal{D}_{t}^{(c)},\mathbf{x}_{j}\in\mathcal{D}_{s}^{(c)} \\ 0,&\text{otherwise}\end{array}\right.\end{array}\right. \tag{4}\]