## 2 Proposed Framework
### Transfer Learning in Manifold Space  
Manifold is homeomorphic spaces in local and Euclidean spaces. It uses Euclidean distance to calculate the distance, which overcomes the feature distortion of transfer learning in original space. The Grassmann manifold can take the original d-dimensional subspace as the basic element to help learning classifier. It usually has an effective numerical form in feature transformation, which can be very efficient representation and solution in the transfer learning problem. In addition, the transfer of source project to target project, or the transfer of source and target projects to a common space are two main methods of feature-based transfer learning. [PERSON] et al. found that the performance of the transfer of source project to target project is better than the latter [19]. In this paper, the optimal subset of source and target projects are transformed into the Grassmann manifold. Geodesic Flow Kernel (GFK) method is used to construct geodesic flow to make the source domain close to the target domain. It integrates the space function of the manifold where these two points are located of the source project with the same distribution as the target project is obtained.  
As shown in the Figure 3, GFK tries to embed the \(D\times d\) dimension subspace \(P_{s}\), \(P_{T}\in R^{D\times d}\) after dimensionless reduction of the source domain and target domain into the manifold G. \(\phi(0)\) is the source domain representation of manifold G, \(\phi(1)\) is the target domain representation of manifold G, and the geodesic flow between \(\phi(0)\) and \(\phi(1)\) is equivalent to transforming the original feature space into an infinite dimension space, reducing the drift phenomenon between domains. The parameterization is shown as follows:  
\[\begin{array}{l}\Phi:t\in[0,1]\rightarrow\Phi:t\in G(d,D)\\ P_{S}~{}=~{}\Phi(0),~{}P_{T}~{}=~{}\Phi(1)\end{array} \tag{16}\]\(\Phi(t)\) is represented as follows, where \(U_{1}\), \(U_{2}\) are orthogonal matrices obtained by SVD.  
\[\Phi(t)\ =\ P_{S}U_{1}\Gamma(t)-R_{S}U_{2}\Sigma(t) \tag{17}\]  
\[P_{S}^{T}P_{T}\ =\ U_{1}\Gamma(t)V^{T},\ R_{S}^{T}P_{T}\ =\ -U_{2}\Sigma(t)V^{T} \tag{18}\]  
\(z^{\infty}\ =\ \left[\Phi(0)^{T}x,\ldots,\Phi(t)^{T}x,\ldots,\Phi(1)^{T}x\right]\) is the feature of manifold space, the inner product of transformed features gives rise to positive semidefinite geodesic flow kernel:  
\[\langle z_{i}^{\infty},z_{j}^{\infty}\rangle\ =\ \int_{0}^{1}\left(\Phi(t)^{T}x_{i} \right)^{T}(\Phi(t)^{T}x_{j})dt=x_{i}^{T}Gx_{j} \tag{19}\]  
Thus, the features in the original space can be transformed into the Grassmann manifold with \(z\ =\ \sqrt{G}x\). \(G\) can be effectively calculated by singular value decomposition.  
\[G\ =\ \left[P_{S}U_{1}\ R_{S}U_{2}\right]\left[\begin{array}{cc}\Lambda_{1}& \Lambda_{2}\\ \Lambda_{2}&\Lambda_{3}\end{array}\right]\left[\begin{array}{c}U_{1}^{T}P_{ S}^{T}\\ U_{2}^{T}R_{S}^{T}\end{array}\right] \tag{20}\]  
Logistic regression is used as classifier to train the source project with the same distribution as the target project, and use the model to distinguish the defect type of the target item module.