## 2 Basic Theory  
### Transfer Component Analysis  
TCA is a transfer learning method based on domain adaptation [23] proposed by [PERSON] et al. [24]. The principle is to utilize the transferability of the features between domains, map the source and target domains to a lower dimensional shared feature space, and search for common transfer components for learning [25]. The mapping process of TCA uses MMD as the metric criterion, which greatly reduces the distribution discrepancy between the source domain and the target domain while retaining its internal attributes so that the edge distribution probability density and conditional probability density of the mapped data source domain and target domain are equal. TCA plays an important role in cross-domain transfer learning, where there is a significant discrepancy in the data distribution between the source and target domains. Equation (1) is the mathematical expression for MMD.  
\[D_{H}(X_{s},X_{t})=\left\|\frac{1}{n_{s}}\sum_{i=1}^{n_{s}}\phi(x_{i})-\frac{1 }{n_{t}}\sum_{j=1}^{n_{t}}\phi(x_{j})\right\|_{H}^{2} \tag{1}\]  
where, \(X_{s}\) and \(X_{t}\) are the sample sets of the source domain and target domain, \(n_{s}\) and \(n_{t}\) are the number of samples in the source domain and target domain, respectively. The variable \(\phi\) is the mapping function that satisfies two probability distributions, \(\phi(x_{i})\) and \(\phi(x_{j})\), minimizing MMD, and \(\left\|\cdot\right\|_{H}^{2}\) is the reproducing kernel Hilbert space (RKHS) norm. If MMD is directly optimized, the mapping function is prone to falling into local optima. Therefore, regularization is introduced to constrain the complexity of the kernel function matrix \(K\) and the mapping matrix \(L\). Equation (1) is simplified as follows:  
\[D_{H}(X_{s},X_{t})=tr(KL) \tag{2}\]  
where \(tr(\cdot)\) is the trace of the matrix. The calculation formula for the kernel function matrix \(K\) is shown in Equation (3), and the calculation for the mapping matrix \(L\) is shown in Equation (4).  
\[K=\begin{bmatrix}K_{s,s}&K_{s,t}\\ K_{t,s}&K_{t,t}\end{bmatrix}\in\mathbb{R}^{(n_{s}+n_{t})\times(n_{s}+n_{t})} \tag{3}\]  
where, \(K_{s,s}\), \(K_{s,t}\), \(K_{t,s}\), \(K_{t,t}\) represent the kernel functions of the source domain, cross domain, and target domain in the mapped space, respectively.  
\[L_{ij}=\begin{cases}\frac{1}{n_{s}^{2}}&x_{i},x_{j}\in X_{s}\\ \frac{1}{n_{t}^{2}}&x_{i},x_{j}\in X_{t}\\ -\frac{1}{n_{s}n_{s}}&\text{otherwise}\end{cases} \tag{4}\]  
To further simplify the calculation, the kernel function matrix \(K\) is mapped to an \(m\)-dimensional space, and a low-dimensional matrix \(W\) is introduced, and the formula is as follows:  
\[\widetilde{K}=(KK^{-1/2}\widetilde{W})(\widetilde{W}^{T}K^{-1/2}K)=KWW^{T}K \tag{5}\]where \(\widetilde{K}\) is a temporary variable, \(W=K^{-1/2}\widetilde{W}\) and \(W^{T}KHKW\) represent the variance of the mapping sample, and \(H=I_{n_{s}+n_{t}}-[1/(n_{s}+n_{t})]II^{T}\) represents the central matrix, \(I\in\mathbb{R}^{(n_{s}+n_{t})\times(n_{s}+n_{t})}\). The ultimate optimization goal of TCA is shown in Equation (6):  
\[\min_{W}=tr\big{(}W^{T}KLKW\big{)}+u\times tr(W^{T}W) \tag{6}\]  
where \(u>0\) is the equilibrium hyperparameter.  
The implementation steps of TCA are as follows: input the feature sample sets \(X_{s}\) and \(X_{t}\) of the source domain and target domain, then select the kernel function to calculate the matrix \(K\), map the matrix \(L\) and matrix \(H\), and finally output the new feature sample sets \(T_{s}\) of the source domain and \(T_{t}\) of the target domain after distribution alignment.