## 3 Methodology
### Kernelization  
The model described in Equation (4) is a linear multi-label classifier, limiting its effectiveness in capturing nonlinear relationships between the input and output. To address this limitation, we propose the utilization of kernel methods to develop nonlinear multi-label classifiers. The kernel function is then introduced into our proposed framework via the dual formulation of the problem in Equation (4) using the Karush-Kuhn-Tucker theorem. Dual variables are then added to transform the constrained problem into an unconstrained problem which can be optimized using the Lagrangian function as shown below:  
\[ \begin{split} L(U,\,V,\,E,\,S,\,w,\,\xi,\,\tau,\,\eta)& =\,\frac{\alpha}{2}||E||_{2,1}+\frac{1}{2}{\sum_{i,j}}\big{|}\big{|}u_{i}-u_{j} \big{|}\big{|}_{2}^{2}s_{ij}\\ &+\gamma||U||_{*}+\,\theta||S||_{*}+\frac{1}{2}{\sum_{p=1}^{l}||w ^{p}||^{2}}\\ &+C{\sum_{i=1}^{N}\frac{1}{|Y_{i}^{+}||Y_{i}^{-}|}{\sum_{p\in Y_{ i}^{+}}\sum_{q\in Y_{i}^{-}}\xi_{pq}^{i}}}\\ &+\frac{u}{2}\bigg{|}\bigg{|}E-X+VU^{T}+\frac{M_{1}}{u}\bigg{|} \bigg{|}_{F}^{2}\\ &-{\sum_{i=1}^{N}\sum_{p\in Y_{i}^{+}}\sum_{q\in Y_{i}^{-}}\tau_ {pq}^{i}\Big{(}<w^{p}-w^{q},u_{i}>+b^{p}-b^{q}-1+\xi_{pq}^{i}\Big{)}}\\ &-{\sum_{i=1}^{N}\sum_{p\in Y_{i}^{+}}\sum_{q\in Y_{i}^{-}}\eta_ {pq}^{i}\xi_{pq}^{i}}\end{split}  \tag{4}\]  
We now seek a saddle point of the Lagrangian, which would be the minimum for the primal variables \(\{w,b,\xi\}\) and the maximum for the dual variable \(\tau\). To find the minimum over the primal variables we require,  
\[ \begin{split}\partial_{b^{p}}L=0&\implies\sum_{i=1 }^{N}\sum_{j\in Y_{i}^{+}}\sum_{q\in Y_{i}^{-}}c_{jq}^{i}\tau_{jq}^{i}=0\\ c_{jq}^{i}=\left\{\begin{array}{ll}0&\text{if $j\
eq p$ and $q\
eq p$,}\\ +1&\text{if $j=p$,}\\ -1&\text{if $q=p$.}\end{array}\right.\end{split}  \tag{5}\]  
Similarly, for \(\xi\) we require  
\[\partial_{\xi_{pq}^{i}}L=0\quad\implies\quad\frac{C}{\big{|}Y_{i}^{+}\big{|} Y_{i}^{-}\big{|}}=\tau_{pq}^{i}+\eta_{pq}^{i} \tag{6}\]  
Similarly, for \(w\) we require  
\[\partial_{w^{p}}L=0\quad\implies\quad w^{p}={\sum_{i=1}^{N}\left(\sum_{j\in Y _{i}^{+}}\sum_{q\in Y_{i}^{-}}c_{jq}^{i}\,\tau_{jq}^{i}\right)}u_{i} \tag{7}\]Let \(\varphi(\cdot)\) be a feature mapping function that maps \(x\) from \(\mathbb{R}^{d}\) to a Hilbert space \(H\). Consequently, the optimization problem described in Equation (4) can be reformulated as follows:  
\[\min\limits_{U,\,V,\,E,\,W,\,S,\,b}\max\limits_{\gamma_{\#}^{i}} \frac{\alpha}{2}||E||_{2.1}+\frac{1}{2}{\sum\limits_{i,j}}||u_{i}-u_{j}||_{2 }^{2}s_{ij}\,+\,\gamma||U||_{*}\] \[+\theta||S||_{*}-\frac{1}{2}{\sum\limits_{p=1}^{l}{\sum\limits_{ h,j=1}^{N}{\beta_{p}^{h}\beta_{p}^{i}\varphi(u_{h})\varphi(u_{i})^{T}}}}\] \[+\sum\limits_{i=1}^{N}{\sum\limits_{p\in Y_{i}^{+}}{\sum\limits_ {q\in Y_{i}^{-}}{\tau_{pq}^{i}}}} \tag{8}\] \[s.t.\quad E=X-VU^{T},\,\,\,\,V^{T}V=I\] \[\text{diag}(S)=0,\,\,\,\,S\geq 0,\,\,\,\,S^{T}1=1\] \[\tau_{jq}^{i}\in[0,C],\,\,\,\,\sum\limits_{i=1}^{N}{\sum\limits_ {j\in Y_{i}^{+}}{\sum\limits_{q\in Y_{i}^{-}}{c_{jq}^{i}\tau_{jq}^{i}}}}=0,for\,\,p= 1,\ldots,l\]  
where \(\beta_{p}^{i}=\sum\limits_{j\in Y_{i}^{+}}{\sum\limits_{q\in Y_{i}^{-}}{c_{ jq}^{i}\tau_{jq}^{i}}}\) and \(w^{p}=\sum\limits_{i=1}^{N}{\beta_{p}^{i}u_{i}}\)  
Define \(K(u_{i},u_{j})=\varphi(u_{i})\varphi(u_{j})^{T}\) to be the kernel matrix (or Gram matrix) in the RKHS.