## 3 Methodology
### Robust Ranking Support Vector Machine  
We commence with the fundamental linear Rank-SVM approach [19] method for multi-label classification. For instance, \(x\in\mathbb{R}^{m}\), its real-valued prediction is obtained by \(f=xW+b\), where \(b=[b_{1},b_{2},\ldots,b_{l}]\in\mathbb{R}^{l}\) is the bias and \(W=\left[w^{1},w^{2},\ldots,w^{l}\right]\in\mathbb{R}^{m\times l}\) is the parameter matrix. To simplify the formulation, we can absorb \(b_{j}\) into \(w^{j}\) by appending \(1\) to each instance \(x\) as an additional feature. The objective of Rank-SVM is to minimize the ranking loss while maximizing the margin. The ranking learning step can be formulated as follows:  
\[ \begin{split}\min_{U,\,V,\,W,\,b}&\frac{1}{2}\sum _{j=1}^{l}\Bigl{|}\Bigl{|}\Bigl{|}w^{j}\Bigr{|}\Bigr{|}^{2}+C\sum_{i=1}^{N} \frac{1}{|Y_{i}^{+}|\bigl{|}Y_{i}^{-}\bigr{|}}\sum_{p\in Y_{i}^{+}}\sum_{q\in Y _{i}^{-}}\hat{\varepsilon}_{pq}^{i}\\ s.t.&<w^{p}-w^{q},u_{i}>+b^{p}-b^{q}\geq 1-\xi_{pq}^{i} \\ (p,q)&\in Y_{i}^{+}\times Y_{i}^{-},\\ \xi_{pq}^{i}&\geq 0,\,\,i=1,\ldots,n.\end{split}  \tag{2}\]  
where \(Y_{i}^{+}\) (or \(Y_{i}^{-}\)) represents the index set of relevant (or irrelevant) labels associated with the instance \(x_{i}\). The notation \(|\cdot|\) represents the cardinality of a set, and the tradeoff hyper-parameter \(C\) is utilized to control the complexity of the model. However, the performance of the Rank-SVM method is very sensitive to noise data samples and may decline sharply when the noise is high. In this paper, the robustness of feature space is used to solve the impact of noise on classification performance. Given a data matrix \(X\in\mathbb{R}^{d\times n}\), where \(d\) represents the feature dimension and \(n\) denotes the number of data samples. To fully study the impact of noise on data, this paper introduces matrix factorization to solve this problem. We decompose the noisy data \(X\) into \(U\) and \(V\), where \(V\in\mathbb{R}^{d\times r}\) is the basis matrix and \(U\in\mathbb{R}^{n\times r}\) is the coefficient matrix, and it can be considered as the new representation of \(X\). Then, training the clean data matrix \(U\) into Rank SVM can make our model more robust and resistant to noise. In addition, since the features selected from the matrix factorization and the multi-label classifier are independent, the low-rank representation is not optimal for the multi-label classification. This paper proposes a new multi-label classification model to improve the classification performance in high noisy data scenarios, seamlessly integrating the matrix factorization and multi-label classifier into a unified framework. Specifically, we use joint learning to combine feature selection and classifier learning, to learn each other's parameters, and finally select the best features through joint optimization, thus realizing a joint learning framework. The robust ranking support vector machine via matrix factorization is as follows:  
\[ \begin{split}\min_{U,\,V,\,E,\,W,\,S,\,b}&\frac{ \alpha}{2}||E||_{2,1}+\frac{1}{2}\!\sum_{i,j}\Bigl{|}\bigl{|}u_{i}-u_{j} \bigr{|}\bigr{|}_{2}^{2}\!s_{ij}\\ &+\gamma||U||_{*}+\theta||S||_{*}+\frac{1}{2}\sum_{p=1}^{l}||w^{ p}||^{2}\\ &+C\!\sum_{i=1}^{N}\frac{1}{|Y_{i}^{+}|\bigl{|}Y_{i}^{-}|}\!\sum_ {p\in Y_{i}^{+}}\sum_{q\in Y_{i}^{-}}\hat{\varepsilon}_{pq}^{i}\\ s.t.& E=X-VU^{T},\,\,\,\,\,V^{T}V=I,\\ &\text{diag}(S)=0,\,\,S\geq 0,\,\,S^{T}\mathbf{1}=1,\\ &<w^{p}-w^{q},u_{i}>+b^{p}-b^{q}\geq 1-\xi_{pq}^{i},\\ &(p,q)\in Y_{i}^{+}\times Y_{i}^{-},\,\xi_{pq}^{i}\geq 0,\,\,i=1, \,\,\ldots,\,n\end{split}  \tag{3}\]where \(S\) is the similarity matrix, in general, a smaller distance \(\big{|}\big{|}u_{i}-u_{j}\big{|}\big{|}\) corresponds to a larger weight \(S_{ij}\), defined as \(\text{diag}(S)=0\) and \(S\geq 0\), which implies \(S_{ii}=0\) and \(S_{ij}\geq 0\). We introduced manifold learning into the proposed framework to maintain manifold relationships between low rank samples to enhance robust low rank representations.