## 3 Methodology
### Optimization  
Since our model involves the \(\ell_{21}\)-norm, the problem's orthogonal and nonnegative constraints are still challenging. Therefore, this article proposes a new and effective optimization method based on ALM to solve the problem. We aim to introduce auxiliary variables to separate constraints and keep their equivalence during optimization. Specifically, we introduce two auxiliary variables \(H=S\) and \(Z=U\) and transform the objective function into the following form  
\[L(U,\,V,\,E,\,S,\,Z,\,H,\,w,\,\xi,\,\tau,\,\eta\,\,M_{1},\,\,M_{2 },\,M_{3})\] \[=\frac{\alpha}{2}||E||_{2.1}+\frac{1}{2}tr\Big{(}U^{T}L_{s}U\Big{)} +\gamma||Z||_{*}\] \[+\frac{u}{2}||E-X+VU^{T}+\frac{M_{1}}{u}||_{F}^{2}+\,\,\frac{u}{ 2}||Z-U+\frac{M_{2}}{u}||_{F}^{2}\] \[+\frac{u}{2}||H-S+\frac{M_{3}}{u}||_{F}^{2}+\theta||H||_{*} \tag{10}\]\[-\frac{1}{2}\sum_{p=1}^{l}\sum_{h,i=1}^{N}\beta_{p}^{h}\beta_{p}^{i }K\big{(}u_{i},u_{j}\big{)}+\sum_{i=1}^{N}\sum_{p\in\{\gamma_{i}^{+}\}}\sum_{q\in \gamma_{i}^{-}}\tau_{pq}^{i}\] \[s.t. \ V^{T}V=I,\ \ \ \text{diag}(S)=0,\ \ S\geq 0,\ \ S^{T}\mathbf{1}=1\] \[\tau_{qq}^{i}\in[0,C],\ \ \ \sum_{i=1}^{N}\sum_{j\in\gamma_{i}^{+}} \sum_{q\in\gamma_{i}^{-}}c_{jq}^{i}\tau_{qq}^{i}=0,\ \ \ \text{for}\ p=1,\ldots,l\]  
where \(u\) is the regularization parameter that determines the penalty for infeasibility, \(M_{1}\in\mathbb{R}^{d\times n}\), \(M_{2}\in\mathbb{R}^{n\times r}\) and \(M_{3}\in\mathbb{R}^{n\times n}\) are the Lagrangian multipliers that penalize the gap between the target and the auxiliary variable. With the transformation, we can adopt alternative optimization to iteratively solve the problem. Specifically, we optimize the objective function with respect to one variable while fixing the remaining variables. The iteration steps are detailed as follows. (1) Update \(U\): By fixing the other variables, the optimization formula for \(U\) becomes  
\[ \begin{split}\min_{U}&\frac{1}{2}\sum_{i,j}\big{|} \big{|}u_{i}-u_{j}\big{|}\big{|}_{2}^{2}S_{ij}+\frac{u}{2}\Big{|}\bigg{|}E-X+VU ^{T}+\frac{M_{1}}{u}\bigg{|}\bigg{|}_{F}^{2}\\ &+\frac{u}{2}\bigg{|}\bigg{|}Z-U+\frac{M_{2}}{u}\bigg{|}\bigg{|}_ {F}^{2}-\frac{1}{2}\sum_{p=1}^{l}\sum_{h,i=1}^{N}\beta_{p}^{h}\beta_{p}^{i}K \big{(}u_{i},u_{j}\big{)}\end{split}  \tag{11}\]  
According to Karush-Kuhn-Tucker (KKT) condition and \(K(U_{i},U_{j})=\exp(-\gamma\big{|}|u_{i}-u_{j}|\big{|}^{2})\), it can be verified that the optimal solution should be  
\[u_{i}=\text{sylvester} \left(\sum_{j=1}^{N}S_{ij}+u+\gamma\sum_{j=1}^{N}K\big{(}U_{i},U_{ j}\big{)}A_{ij},\,UV^{T}V,\right.\] \[\left.\sum_{j=1}^{N}u_{j}S_{ij}-U\left(E_{i:}^{T}-X_{i:}^{T}+\frac {M_{1_{k}}^{T}}{U}\right)\,V\right.\] \[\left.+U\left(Z_{i:}+\frac{M_{2_{i}}}{U}\right)+\gamma\sum_{j=1} ^{N}K\big{(}U_{i},U_{j}\big{)}\big{(}U_{j}\big{)}A_{ij}\right) \tag{12}\]  
(2) Update \(V\): By fixing the other variables, the optimization formula for \(V\) becomes  
\[\min_{V}\ \frac{u}{2}\bigg{|}\bigg{|}E-X+VU^{T}+\frac{M_{1}}{u}\bigg{|}\bigg{|}_ {F}^{2} \tag{13}\]  
Considering \(V^{T}V=I\), the above formula can be rewritten as  
\[\min_{V}\ \bigg{|}\bigg{|}VU^{T}-\left(X-E-\frac{M_{1}}{U}\right)\bigg{|}\bigg{|}_ {\mathbf{F}}^{2} \tag{14}\]  
This problem is commonly referred to as the orthogonal procrustes problem, for which the global optimal solution can be obtained through the singular value decomposition of \(U^{T}\Big{(}X-E-\frac{M_{1}}{u}\Big{)}^{T}\).