## 3 Proposed Framework
### Reducing Cross-Domain Discrepancy Based on Sinkhorn Distance  
Similarity measurement is crucial for evaluating the degree of similarity between feature vectors. For feature vectors with ample representational capabilities, the choice of a similarity measurement algorithm directly impacts classification accuracy [52]. In metric learning, commonly used algorithms include Euclidean distance and cosine distance. Euclidean distance emphasizes absolute differences in numerical values between vectors, making it sensitive to outliers [53]. Cosine distance, on the other hand, focuses on the relative differences in the direction of vectors, remaining insensitive to absolute values [54]. However, cross-domain text features are prone to noise, category-independent feature points, and feature distribution that fails to meet the assumption of independent identical distribution. In such cases, using Euclidean or Cosine distance is inadequate. To address this, we employ the Box-Cox transformation to redistribute feature vectors extracted by mBERT into Gaussian or quasi-Gaussian distributions [55]. We then use the Sinkhorn distance as the similarity measurement for feature vectors [56]. Sinkhorn distance, proposed by [PERSON], based on optimal transport, differs from traditional similarity measurement algorithms like Euclidean and cosine distances [57]. It calculates the work required to move from one vector to another, offering a unique perspective compared to direct physical distance calculations. The Sinkhorn distance can be seen as an optimal mapping from the source domain distribution to the target domain distribution, aiding in the understanding of inter-domain differences while offering greater stability compared to traditional metrics like Maximum Mean Discrepancy (MMD) that may be susceptible to local optima. It effectively handles high-dimensional, nonlinear data distribution disparities. This capability is particularly crucial in natural language processing, where text data often exhibit intricate semantic and syntactic structures, challenging for conventional distance measures to capture. Moreover, the Sinkhorn distance serves not only as a measure between domains but also supports domain adaptation learning. Minimizing the Sinkhorn distance between source and target domains facilitates learning more generalized feature representations, thereby enhancing model performance on the target domain. Importantly, grounded in optimal transport theory, the Sinkhorn distance provides robust theoretical underpinnings and mathematical guarantees. Under suitable conditions, it ensures convergence to genuine distribution disparities, a critical factor in several practical applications. The computation process of Sinkhorn distance is illustrated in Figure 3, where \(P(x)\) and \(Q(x)\) represent two vectors, and the arrows depict the movement from \(P(x)\) to \(Q(x)\). Figure 3: Illustration of Sinkhorn distance. The aforementioned optimal transport process is defined as follows:  
\[d(a,b)=\sum_{s\in U(a,b)}<S,L>\;=\;\sum_{i,j}S_{ij}L_{i,j} \tag{4}\]  
where \(S\) represents a coupling matrix. \(a\) and \(b\) are probability distributions of vectors \(P\) and \(Q\), respectively, ensuring that the sums of elements in \(a\) and \(b\) are both 1. \(U(a,b)=\big{\{}S\in R_{+}^{n\times m}\big{\}}\) denotes that \(U\) is an \(n\times m\) matrix with rows summing to \(a\) and columns summing to \(b\). \(L_{ij}\) signifies the cost of moving from point \(i\) to point \(j\). Since \(S\) is a probability distribution matrix, while Sinkhorn distance holds a natural advantage when the vector distributions are similar or approximately similar, complexities arise in computations, particularly with higher-dimensional vectors and extensive data.