## 2 Materials and Methods
### Methods
#### 2.3.1 Related Work
\[A=M_{s}^{-1/2},B=M_{t}^{-1/2} \tag{2}\]1  
where \(\Phi\) represents the geodesic flow; \(x\) and \(z\) are tangent space features and newly learned Grassmann manifold, respectively; \(G=\Phi(t)\Phi(t)^{T}\) is the transformation from \(x\) to \(z\). Using the transformation, one receives the Grassmann manifold features in (12). \[A=M_{s}^{-1/2},B=M_{t}^{-1/2} \tag{2}\]2  
The MMFT classifier is implemented in the fourth phase of the MMFT framework. The structural risk (13) is minimized across source domains, and conditional alignment is summarized, mainly to train the prediction model [27]. \[A=M_{s}^{-1/2},B=M_{t}^{-1/2} \tag{2}\]3  
where \(E\) is the diagonal label indicator matrix. \[A=M_{s}^{-1/2},B=M_{t}^{-1/2} \tag{2}\]4  
Equation (14) signifies a conditional distribution discrepancy of a prediction model denoted by \(f\). As such, the conditional distribution alignment is defined by \(D_{f,(c)}(Q_{s},Q_{t})\). \[A=M_{s}^{-1/2},B=M_{t}^{-1/2} \tag{2}\]5  
Equation (15) represents a classifier denoted by \(f\) obtained by combining (13) and (14) with both \(D_{s}\) and \(D_{t}\) representing source and target domain samples, respectively [24]. \[A=M_{s}^{-1/2},B=M_{t}^{-1/2} \tag{2}\]6  
Equation (16) accepts an expansion when the representer theorem is implemented with coefficients vector denoted by \(\alpha\), while estimation of an original feature vector to Hilbert space through feature mapping generates a kernel denoted by \(K\)[24]. \[A=M_{s}^{-1/2},B=M_{t}^{-1/2} \tag{2}\]7  
Equation (17) defines a structural risk minimization on the source domain. In this case, \(\alpha\) represents the coefficient vector, while the frobenious norm is denoted by \(\|\ \|_{K}^{2}\), and the kernel matrix is denoted by \(K\in{}^{nxn}\). Both source and target domain label matrices are denoted by \(Y=[y_{1}\ldots y_{n}]\), while the trace operation is represented by \(tr()\)[24]. \[A=M_{s}^{-1/2},B=M_{t}^{-1/2} \tag{2}\]8  
To further transform the conditional distribution alignment, the representer theorem is combined with the kernel to produce (18), with the maximum mean discrepancy matrix denoted by \(M_{c}\)[24]. \[A=M_{s}^{-1/2},B=M_{t}^{-1/2} \tag{2}\]9  
Furthermore, the weighted MMFT (w-MMFT) algorithm is also explored to address the challenge of class imbalance, which occur as a result of source domain label information, that tends to affect the structural risk function used to train the MMFT classifier [24; 28]. In this instance, w-MMFT's objective function is defined by (23), with \(W\) representing the weight matrix. \[A=M_{t}^{1/2}M_{s}^{-1/2},B=I \tag{3}\]0  
Equation (24) defines the solution to the objective function denoted by (23), with \(M_{c}\) representing a maximum mean discrepancy matrix, while \(W\) represents the weight matrix [24]. The overall procedure of MMFT is summarized in Algorithm 1. ```
0: source domains samples \(\left\{\left\{\mathbf{X}_{\mathbf{s}_{\mathbf{s}_{\mathbf{s}}}^{(j)},y_{\mathbf{s}_{\mathbf{s}}}^{ \mathbf{n}_{\mathbf{s}}}}\right\}_{j=1}^{2}\right\}_{j=1}^{2}\); target domain samples \(\left\{\mathbf{X}_{\mathbf{t}_{\mathbf{s}}}\right\}_{i=1}^{\mathbf{n}_{\mathbf{t}}}\); regularization parameters \(\lambda,\eta\); number of iterations \(N\). 0: Predict labels \(\widehat{y}_{t}\) for target domain
1: Calculate the covariance matrices for both \(z\) source domains and target domain, apply DMA, get aligned covariance matrices \(\left\{\widehat{P}_{\mathbf{s}}^{(j)}\right\}_{j=1}^{2}\) and \(\widetilde{P}_{\mathbf{t}}\). 2: Get tangent space features \(\widehat{X}\) form SPD manifold. 3: Learn Grassmann manifold features \(Z=\left[Z_{t},Z_{S}^{1},\ldots,Z_{S}^{2}\right]\) for \(z\) source domains and target domain. 4: Pseudo labels for target domain \(\hat{y}_{t}=[]\). 5: for \(n=1,\ldots,N\) do
6: Multi-source classifier \(f=[]\). 7: for \(j=1,\ldots,z\) do
8: Construct kernel K using features \(\left[Z_{t},Z_{S}^{(j)}\right]\).