## 3 Methods
The objective function of RNMF can be formulated as follows:  
\[\min_{W,H}\parallel X-WH\parallel_{2,1}\text{ s. t. }W\geqslant 0,H\geqslant 0. \tag{4}\]  
The definition of L\({}_{2,1}\) norm is  
\[\parallel X\parallel_{2,1}=\sum_{i}^{n}\sqrt{\sum_{j}^{m}X_{ji}^{2}}=\sum_{i} ^{n}\parallel x_{i}\parallel \tag{5}\]  
where \(\parallel\cdot\parallel_{2,1}\) refers to the L\({}_{2,1}\) norm, and \(x_{i}\) represents the \(i_{th}\) vector of \(X\). Based on the theory of manifold learning [30], manifold regularization can reduce the complexity of data in high-dimensional space while preserving the proximity relationship between neighboring data points in the low-dimensional manifold space. This implies that it is beneficial for dimensionality reduction and feature selection in the low-dimensional manifold space. By projecting the data onto the low-dimensional manifold space, we can reduce the dimensionality of the data and retain the important information that reaches these low-dimensional spaces. Through manifold regularization, it is possible to differentiate adversarial perturbation information, which helps eliminate adversarial perturbation in reconstruction. The objective function of KRMNMF is defined as follows:  
\[ \begin{array}{l}\min_{F,H}\parallel\phi(X)-\phi(X)FH\parallel_{2,1}+\lambda \text{Tr}\big{(}HLH^{\text{T}}\big{)}\\ \text{s. t. }F\geqslant 0,H\geqslant 0\end{array}  \tag{6}\]  
where \(\in R_{+}^{m\times n},W\in R_{+}^{m\times k},H\in R_{+}^{k\times n}\), \(\phi\) is the Gaussian kernel function mapping, \(x_{i}\rightarrow\phi(x_{i})\), \(x_{i}\) represents the \(i_{th}\) sample point in the original data space \(X\), i.e., \(X\rightarrow\phi(X)\); \(\lambda\) is a non-negative regularization parameter, \(\text{Tr}\ (\cdot)\) denotes the trace of a matrix, and \(L\) is the Laplacian matrix defined as \(L=V-W\), where \(V\) is a diagonal matrix and \(V_{ii}=\sum_{j}W_{ij}\). The objective function of KRMNMF is not jointly convex with respect to \(F\) and \(H\). However, when fixing the other variables, the individual variables are convex. In this paper, the Lagrangian multiplier method is adopted to solve the problem. The equivalent form of the objective function is as follows:  
\[\min_{F,H}\text{Tr}\Big{(}(\phi(X)-\phi(X)FH)D(\phi(X)-\phi(X)FH)^{\text{T}} \Big{)}+\lambda\text{Tr}\Big{(}HLH^{\text{T}}\Big{)} \tag{7}\]  
where \(D\) is a diagonal matrix that satisfies  
\[D_{ii}=\left(\sum_{j=1}^{n}\left(\phi(X)-\phi(X)FH\right)_{ji}^{2}\right)^{- \frac{1}{2}} \tag{8}\]Let \(\phi(x_{i})^{\rm T}\phi(x_{i})=K\); therefore,  
\[D_{ii}=\left[({\rm I}-FH)^{\rm T}K({\rm I}-FH)\right]_{ii}^{-\frac{1}{2}} \tag{9}\]  
where I represents the identity matrix. Equation (7) is equivalent to  
\[ \begin{array}{l}\underset{F,H}{\rm min}{\rm Tr}\Big{(}\phi(X)D_{\phi}(X)^{ \rm T}\Big{)}-2{\rm Tr}\Big{(}\phi(X)DH^{\rm T}F^{\rm T}\phi(X)^{\rm T}\Big{)}+ \\ {\rm Tr}\Big{(}\phi(X)FHDH^{\rm T}F^{\rm T}\phi(X)^{\rm T}\Big{)}+\lambda\big{(} HLH^{\rm T}\big{)}\end{array} . \tag{10}\]  
Let \(\Psi\) and \(\Omega\) be the Lagrange multipliers for \(F\) and \(H\), respectively. The Lagrangian function with respect to the objective function is given by the following:  
\[ \begin{array}{l}L=\underset{F,H}{\rm min}{\rm Tr}\Big{(}\phi(X)D\phi(X)^{ \rm T}\Big{)}-2{\rm Tr}\Big{(}\phi(X)DH^{\rm T}F^{\rm T}\phi(X)^{\rm T}\Big{)} +\\ {\rm Tr}\Big{(}\phi(X)FHDH^{\rm T}F^{\rm T}\phi(X)^{\rm T}\Big{)}+\lambda\big{(} HLH^{\rm T}\big{)}+{\rm Tr}\Big{(}\Psi^{\rm T}F\Big{)}+{\rm Tr}\Big{(}\Omega^{ \rm T}H\Big{)}.\end{array}