# Fuzzy Graph Learning Regularized Sparse Filtering for Visual Domain Adaptation
## 3 Methodology
### Fuzzy Graph Learning for Domain Alignment  
How to reduce the domain discrepancy is a key problem for DA, moment-based and adversarial training-based strategies cannot guarantee the separability of target samples, \begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Description** \\ \hline \(\mathcal{D}_{s}/\mathcal{D}_{t}\) & source/target domain \\ \(x_{s}/x_{t}\) & original source/target domain data \\ \(y_{s}/y_{t}\) & source/target domain label \\ \(n_{s}/n_{t}\) & number of source/target samples \\ \(m/k\) & original/transformed feature dimension \\ \(\tilde{x}_{s}/\tilde{x}_{t}\) & source/target features \\ \(f(\cdot)\) & mapping function, \(\tilde{X}=f(X)\) \\ \(W\) & the transformation matrix to be solved \\ \(L_{FGL}\) & objective function for fuzzy graph learning \\ \(L_{SF}\) & objective function for sparse filtering \\ \(\lambda\) & the balance factors between two objectives \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations and descriptions used in this paper. because they do not exploit the structural information existing across different classes. Therefore, we propose fuzzy graph learning for class-wise alignment. The underlying idea is: samples belonging to the same class (no matter which domain it comes from) should be close to each other. Since there are no labels for target samples, we must adopt pseudo labels as a substitution. Obviously, the quality of pseudo labels will greatly influence the learned classifier/representation, it may cause a catastrophic accumulation of errors if not handled properly. For our method, we propose fuzzy pseudo labels which employ a probability matrix to update the classifier. In Figure 1, we give a graphical illustration to explain how soft label works. The class centroids are calculated by labeled source samples and the decision boundary can be obtained by means of the distance to each centroid. For an unreliable training sample which located in the intersection of decision boundary (also can be described as it has similar distance to each centroid), the probabilities of it belonging to any category is extremely close, e.g., 24%, 25%, 26%. In such situation, assigning the label with biggest probability is risky. Like the figure in the middle shows, the updated centroid would have a big influence to decision boundaries. When we adopt soft labels, the case is shown in the last figure, each centroid would move a distance to the training sample according to corresponding probability value. Intuitively, soft labels have significant less effect on decision boundaries than hard. Specifically, suppose we have the learned representation \(\hat{x_{s}}=f(x_{s})\in R^{k\times n_{s}}\) and \(\hat{x_{t}}=f(x_{t})\in R^{k\times n_{t}}\) (we will introduce the transformation \(f(\cdot)\) later), along with the source labels \(y_{s}\in[1,2,\cdots,c]\). We use \(x=[\hat{x_{s}},\hat{x_{t}}]\in R^{k\times(n_{s}+n_{t})}\) to denote the combination of two domains' features. The objective of fuzzy graph learning can be written as:  
\[L_{\text{FGL}}=\frac{1}{2(n_{s}+n_{t})}\sum_{i=1}^{n_{s}+n_{t}}\sum_{j=1}^{n_{s }+n_{t}}(x_{i}-x_{j})^{2}W_{ij} \tag{4}\]  
where \(W\in R^{(n_{s}+n_{t})\times(n_{s}+n_{t})}\) is the affinity graph where \(W_{ij}\in[0,1]\) denotes the similarity between \(x_{i}\) and \(x_{j}\), and \(W\) can be defined as follows:  
\[W= \begin{pmatrix}W_{ss}&W_{st}\\ W_{ts}&W_{tt}\end{pmatrix}  \tag{5}\]  
Fuzzy connections only exist in cross-domain cases, so we describe how to construct the affinity graph separately:  
Figure 1: Unreliable training samples and its influence on decision boundary. Construction of \(W^{ss}\): For labeled source domain, we adopt the supervised graph, that is, the edges exist only between samples with the same label. Ideally, this can help obtain discriminative representations. \[W^{ss}_{ij}= \begin{cases}1,&if\ Y_{i}=Y_{j}\\ 0,&otherwise\end{cases}  \tag{6}\]  
Construction of \(W^{tt}\): For unlabeled target domain, we encourage target samples to be close to their neighborhood samples, thus a cut-off is necessary.