## 3 Methodology
### Optimization
To be more specific, given  
\[[V_{1}\ S_{1}\ U_{1}]=\text{SVD}\Bigg{(}U^{T}\bigg{(}X-E-\frac{M_{1}}{U} \bigg{)}^{T}\Bigg{)} \tag{15}\]  
The following formula can update \(V\)  
\[V=U_{1}{V_{1}}^{T} \tag{16}\](3) Update \(E\): By fixing the other variables, the optimization formula for \(E\) becomes  
\[\min_{E}\ \frac{\alpha}{2}||E||_{2.1}+\frac{U}{2}\bigg{|}\bigg{|}E-\bigg{(}X-VU^{T}- \frac{M_{1}}{U}\bigg{)}\bigg{|}\bigg{|}_{\mathbf{F}}^{2} \tag{17}\]  
Let \(T=\Big{(}X-VU^{T}-\frac{M_{1}}{u}\Big{)}\), \(\tau=\frac{\alpha}{2}\), and we further have  
\[\min_{E}\ \tau||E||_{2.1}+\frac{1}{2}||E-T||_{\mathbf{F}}^{2} \tag{18}\]  
To solve the above equation, we introduce the following lemma, also presented in [28], with detailed proof. **Lemma 1**. : _Given a matrix \(W=\{w_{1},w_{2},\ldots,w_{n}\}\in\mathbb{R}^{m\times n}\) and a positive scalar \(\lambda\), then \(X^{*}\) is the optimal solution of_  
\[\min_{E}\ \lambda||X||_{2.1}+\frac{1}{2}||X-W||_{\mathbf{F}}^{2} \tag{19}\]  
_and the column of \(X^{*}\)_  
\[X^{*}_{i}=\left\{ \begin{array}{cc}\frac{\|W_{i}\|-\lambda}{\|W_{i}\|}W_{i} \,&\text{if}\ \lambda<\|W_{i}\|\\ 0\ \ \ \,&\text{otherwise}\end{array} \right. \tag{20}\]  
According to Lemma 1, the solution of the above problem is  
\[E(\cdot,\ i)=\left\{ \begin{array}{cc}\frac{\|T\|_{\mathbf{F}}^{-\tau}}{\|T \|_{\mathbf{i}}^{2}}T_{i}\,&\text{if}\ \tau<\|T_{i}\|\\ 0\ \ \ \,&\text{otherwise}\end{array} \right. \tag{21}\]  
(4) Update \(Z\): By fixing the other variables, the optimization formula for \(Z\) becomes  
\[\min_{Z}\ \gamma||Z||_{*}+\ \frac{u}{2}\bigg{|}\bigg{|}Z-U+\frac{M_{2}}{u} \bigg{|}\bigg{|}_{F}^{2} \tag{22}\]  
We first introduce the soft thresholding (shrinkage) operator  
\[S_{\epsilon}[q]=\max(|q|-\epsilon,0)\text{sgn}(q) \tag{23}\]  
where \(\text{sgn}(q)\) represents the \(sign\) function. By applying the shrinkage operator element-wise to the singular values of \(U-\frac{M_{2}}{u}\), the optimal update for \(Z\) can be expressed as follows:  
\[Z=U_{1}S_{\frac{u}{2}}[S_{1}]V_{1}^{T} \tag{24}\]  
in which \(U_{1}\), \(S_{1}\) and \(V_{1}\) are the SVD factorization  
\[\Big{[}U_{1},S_{1},V_{1}^{T}\Big{]}=\text{SVD}\bigg{(}U-\frac{M_{2}}{u}\bigg{)} \tag{25}\]  
(5) Update \(S\): By fixing the other variables, the optimization formula for \(S\) can be derived as  
\[\min_{S}\ \frac{1}{2}\sum_{i,j}||U_{i}-U_{j}||_{2}^{2}S_{ij}+\frac{u}{2 }\bigg{|}\bigg{|}H-S+\frac{M_{3}}{u}\bigg{|}\bigg{|}_{F}^{2} \tag{26}\] \[s.t.\ \ \ \text{diag}(S)=0,S\geq 0,S^{T}\mathbf{1}=1\]  
where \(\big{\|}U_{i}-U_{j}\big{\|}_{2}^{2}=G_{ij}\), then  
\[\min_{S}\ \frac{1}{2}\text{Tr}\Big{(}G^{T}S\Big{)}+\frac{\mu}{2}\bigg{\|}S- \bigg{(}H+\frac{M_{3}}{u}\bigg{)}\bigg{\|}_{F}^{2} \tag{27}\]We denote \(N=H-\frac{M_{3}}{\mu}\)  
\[ \begin{split}&\underset{S_{i}}{\min}\|S_{i}-(N_{i}-G_{i}/2\mu)\|_{2 }^{2}\\ & s.t.\ \ \ \text{diag}(S)=0,S\geq 0,S^{T}\mathbf{1}=1\end{split}  \tag{28}\]  
Considering the constraints  
\[ \begin{split}&\underset{S_{i}}{\min}\underset{s}{\sum}\|S_{i}-(N_{i}-G_{i }/2\mu)\|_{2}^{2}-\eta\left(1^{T}S_{i}-1\right)-\zeta^{T}S_{i}\\ & s.t.\ \ \text{diag}(S)=0\end{split}  \tag{29}\]  
Taking the derivative with respect to \(S_{i}\) and setting it to zero  
\[S_{i}-(N_{i}-G_{i}/2\mu)-\eta 1-\zeta=0 \tag{30}\]  
The \(j\) entry of \(S_{i}\) is shown below  
\[S_{ij}-\left(N_{ij}-G_{ij}/2^{\tau}\right)-\eta-\zeta_{i}=0 \tag{31}\]  
According to KKT conditions  
\[S_{ij}=\left(\left(N_{ij}-G_{ij}/2\mu\right)+\eta\right)_{+} \tag{32}\]  
(6) Update \(H\): By fixing the other variables, the optimization formula for \(H\) becomes  
\[\underset{H}{\min}\ \ \theta||H||_{*}+\frac{u}{2}\left|\left|H-S+\frac{M_{3}}{u }\right|\right|_{\mathbf{F}}^{2} \tag{33}\]  
To begin, we introduce the soft thresholding (shrinkage) operator  
\[S_{\varepsilon}[q]=\max(|q|-\varepsilon,0)\text{sgn}(q) \tag{34}\]  
where \(\text{sgn}(q)\) represents the \(sign\) function.