## Proposed Proportionate NMCC Algorithms
### Proportionate NMCC with a CIM  
Here, we first introduce the CIM theory [41; 42; 43; 44; 45; 46; 47; 48; 49; 50]. As we know, the correntropy method is achieved from nonlinear measurement of random similar vectors \(\mathbf{g}=[ \begin{array}{cccc}g_{1}&g_{2}&\cdots&g_{N}\end{array} ]^{T}\) and \(\mathbf{f}=[ \begin{array}{cccc}f_{1}&f_{2}&\cdots&f_{N}\end{array} ]^{T}\), which are in kernel space. Thus, we have  
\[L(\mathbf{g},\mathbf{f})=\mathbf{E}[\kappa(\mathbf{g},\mathbf{f})]=\int\kappa (\mathbf{g},\mathbf{f})dF_{gf}(g,f), \tag{15}\]where \(\kappa(\mathbf{g},\mathbf{f})\) represents the shift-invariant Mercer kernel, and \(F_{gf}(g,f)\) represents the joint distribution of \(\mathbf{g}\) and \(\mathbf{f}\), and \(\mathbf{E}[\mathbf{x}]\) is the expectation operation of \(\mathbf{x}\). In general, the distribution of the data is usually unknown. Moreover, only a few samples of \(\{g_{i},f_{i}\}\) are available in practice. Thus, the correntropy can be obtained by [41; 42; 43; 44; 45; 46; 47; 48; 49; 50]  
\[\bar{L}(\mathbf{g},\mathbf{f})=\frac{1}{N}\sum_{i=1}^{N}\kappa(g_{i},f_{i}). \tag{16}\]  
According to the correntropy theory, the Gaussian kernel is a kind of typical kernel, which is given by [41; 42; 43; 44; 45; 46; 47; 48; 49; 50]  
\[\kappa(g,f)=\frac{1}{\sigma_{1}\sqrt{2\pi}}\exp\left(-\frac{e^{2}}{2\sigma_{1} ^{2}}\right), \tag{17}\]  
where \(e=g-f\), and \(\sigma_{1}\) is the width of a kernel. The used correntropy has a boundary under various distributions and it can give robustness characteristics for non-Gaussian noise, which has been used for exploiting MCC algorithms. Next, we give the CIM to better understand the correntropy theory. By considering \(\mathbf{g}\) and \(\mathbf{f}\) in a sampling space, we can define the CIM as [41; 42; 43; 44; 45; 46; 47; 48; 49; 50]  
\[\text{CIM}(\mathbf{g},\mathbf{f})=\sqrt{\left(\kappa(0)-\bar{L}(\mathbf{g}, \mathbf{f})\right)}, \tag{18}\]  
where \(\kappa(0)=\frac{1}{\sigma_{1}\sqrt{2\pi}}\). Thus, we can get  
\[\text{CIM}^{2}(\mathbf{g},0)=\frac{1}{\sigma_{1}N\sqrt{2\pi}}\sum_{i=1}^{N} \left(1-\exp\left(-\frac{a_{i}^{2}}{2\sigma_{1}^{2}}\right)\right). \tag{19}\]  
It can be seen that, if \(|g_{i}|>\varsigma\), \(\forall g_{i}\
eq 0\), then, as \(\sigma_{1}=0\), we found that the CIM is an \(l_{0}\)-norm approximation, and \(\varsigma>0\) is set to be a very small constant. Finally, we propose a CIM penalized PNMCC (CIM-PNMCC) algorithm, which is realized by introducing CIM penalty into PNMCC's cost function to construct a sparse PNMSS with a implementation of the CIM. We also take the gain controlling matrix into consideration to design the desired zero-attractor. According to the zero attracting sparse AF algorithms, the developed CIM-PNMCC is to calculate  
\[ \begin{split}&(\mathbf{\tilde{w}}(n+1)-\mathbf{\tilde{w}}(n))^{ \mathsf{T}}\mathbf{G}^{-1}(n)(\mathbf{\tilde{w}}(n+1)-\mathbf{\tilde{w}}(n)) +\gamma_{\text{CIM}}\mathbf{G}^{-1}(n)\text{CIM}^{2}(\mathbf{\tilde{w}}(n+1),0)\\ &\text{subject to }\hat{e}(n)=\left[1-\varsigma\exp\left(-\frac{e ^{2}(n)}{2\sigma^{2}}\right)\right]e(n),\end{split}  \tag{20}\]  
where \(\mathbf{G}^{-1}(n)\) denotes the inverse operation of \(\mathbf{G}(n)\). \(\gamma_{\text{CIM}}>0\) is a regularization, which is a balance between the system identification errors and CIM penalty of \(\mathbf{\tilde{w}}(n+1)\). Our devised CIM-PNMCC is different from the general zero attracting since it provides a scaling by \(\mathbf{G}^{-1}(n)\) on the CIM penalty. To get the minimization of (20), the LM method is employed. Therefore, the proposed CIM-PNMCC's cost function is  
\[ \begin{split}& J(n+1)=(\mathbf{\tilde{w}}(n+1)-\mathbf{\tilde{w}}(n))^{ \mathsf{T}}\mathbf{G}^{-1}(n)(\mathbf{\tilde{w}}(n+1)-\mathbf{\tilde{w}}(n)) +\gamma_{\text{CIM}}\mathbf{G}^{-1}(n)\text{CIM}^{2}(\mathbf{\tilde{w}}(n+1),0)\\ &+\lambda\left(\hat{e}(n)-\left[1-\varsigma\exp\left(-\frac{e^{2} (n)}{2\sigma^{2}}\right)\right]e(n)\right).\end{split}