## 2 Related Work and Methods
### Proposed Method
#### 2.4.3 Optimization Method  
\(\ell_{2,1/2}\)-norm possesses non-convex and non-Lipschitz continuous properties, and there is no universal method for solving such problems. In this paper, we still employ the commonly used gradient descent method in neural networks to calculate the optimization solution of the objective function. The specific form of the objective function is as follows:  
\[L(W_{1};W_{2};b_{1};b_{2})=J(X;\hat{X})+\lambda(\|W_{1}\|_{2,\frac{1}{2}}^{ \frac{1}{2}}+\|W_{2}\|_{2,1})+\beta\Big{(}\text{Tr}(YL^{n}Y^{T})+(\text{Tr}(W _{1}L^{d}W_{1}^{T})\Big{)}\]  
\[=\frac{1}{n}\|\hat{X}-X\|_{2,\frac{1}{2}}^{\frac{1}{2}}+\lambda(\|W_{1}\|_{2, \frac{1}{2}}^{\frac{1}{2}}+\|W_{2}\|_{2,1})+\beta\text{Tr}(YL^{n}Y^{T})+\alpha \text{Tr}(W_{1}L^{d}W_{1}^{T}) \tag{17}\]  
The weight matrix \(W_{1}\) is updated using gradient descent as follows:  
\[W_{1}^{(t)}=W_{1}^{(t-1)}-\gamma\times\
abla_{W_{1}^{(t-1)}}L(W_{1};W_{2};b_{1 };b_{2}) \tag{18}\]  
where \(W_{1}^{(t)}\) represents the t iteration of \(W_{1}\), \(\gamma\) represents the learning rate of the autoencoder, and \(\
abla_{W_{1}^{(t-1)}}L(W_{1};W_{2})\) represents the gradient of \(W_{1}\). Furthermore, the gradient of \(W_{1}\) is computed as follows:  
\[ \begin{array}{l}\
abla_{W_{1}^{(t-1)}}L(W_{1};W_{2};b_{1};b_{2})=\frac{ \partial L(W_{1}W_{2};b_{1};b_{2})}{\partial W_{1}}\\ =\frac{\partial J(W_{1};W_{2};b_{1};b_{2})}{\partial W_{1}}+\lambda W_{1}D+ \beta\frac{\partial\text{Tr}(YL^{n}Y^{T})}{\partial W_{1}}+\alpha\frac{ \partial\text{Tr}(W_{1}L^{d}W_{1}^{T})}{\partial W_{1}}\end{array}  \tag{19}\]  
where \(D\) is a diagonal matrix:  
\[D=diag\left\{\frac{1}{2\times\left|\left|w_{1}^{(1)}\right|_{2}^{\frac{1}{2}} ^{\frac{1}{2}},\frac{1}{2\times\left|\left|w_{2}^{(1)}\right|_{2}^{\frac{1}{2} },\cdots,\frac{1}{2\times\left|\left|w_{m}^{(1)}\right|_{2}^{\frac{1}{2}} \right.}\right\}\right.\right.\right. \tag{20}\]  
where \(w_{m}^{(1)}\) denotes the \(m\)-th row of \(W_{1}\). The derivative of the loss function \(J(W_{1};W_{2};b_{1};b_{2})\) is as follows:  
\[\frac{\partial J(W_{1};W_{2};b_{1};b_{2})}{\partial W_{1}}=\frac{\partial \left|\left|\hat{X}-X\right|\right|_{2,\frac{1}{2}}^{\frac{1}{2}}}{\partial W_ {1}}\]  
\[=\frac{1}{n}(W_{2}^{T}(\frac{\hat{X}-X}{2\left|\left|\hat{X}-X\right|\right|_{2 }^{3/2}}\cdot\hat{X}\cdot[1-\hat{X}]))\cdot Y\cdot[1-Y]X^{T} \tag{21}\]  
where the element-wise product operator is indicated by \(\cdot\), the activation function \(\sigma\) is the sigmoid function \(Y=(W_{1}+b_{1})\). \[\frac{\partial\text{Tr}(YL^{n}Y^{T})}{\partial W_{1}}=2 YL^{n}\cdot Y\cdot(1-Y )X^{T}\]\[\frac{\partial Tr(W_{1}L^{d}W_{1}{}^{T})}{\partial W_{1}}=2W_{1}L^{d} \tag{22}\]  
In a similar vein, iterative calculation yields the weight matrix \(W_{2}\):  
\[W_{2}^{(t)}=W_{2}^{(t-1)}-\gamma\times\
abla_{W_{2}{}^{(t-1)}}L(W_{1};W_{2};)=W _{2}^{(t-1)}-\gamma\times\frac{\partial L(W_{1};W_{2};b_{1};b_{2})}{\partial W_ {2}} \tag{23}\]  
\[\frac{\partial L(W_{1};W_{2};b_{1};b_{2})}{\partial W_{2}}=\frac{\partial J(W_ {1};W_{2};b_{1};b_{2})}{\partial W_{2}}+\lambda BW_{2} \tag{24}\]  
\[B=diag\left\{\frac{1}{\left|\left|w_{1}^{(2)}\right|\right|_{2}},\frac{1}{ \left|\left|w_{2}^{(2)}\right|\right|_{2}},\dots,\frac{1}{\left|\left|w_{d}^{( 2)}\right|\right|_{2}}\right\} \tag{25}\]  
where \(w_{d}^{(1)}\) denotes the \(d\)-th row of \(W_{2}\). In a similar way, \(b_{1}\) and \(b_{2}\) iterations are calculated as follows:  
\[b_{1}^{(t)}=b_{1}^{(t-1)}-\gamma\times\
abla_{b_{1}{}^{(t-1)}}L(W_{1};W_{2};b_ {1};b_{2})=b_{1}^{(t-1)}-\gamma\times\frac{\partial L(W_{1};W_{2};b_{1};b_{2}) }{\partial b_{1}} \tag{26}\]  
\[b_{2}^{(t)}=b_{2}^{(t-1)}-\gamma\times\
abla_{b_{2}{}^{(t-1)}}L(W_{1};W_{2};b_ {1};b_{2})=b_{2}^{(t-1)}-\gamma\times\frac{\partial L(W_{1};W_{2};b_{1};b_{2}) }{\partial b_{2}} \tag{27}\]  
According to the updating rules for DGA presented above, the overall optimization process is given in Algorithm 1.