## Appendix A
\begin{array}{l}\mathbf{H}\{\ln[\sigma_{m}{}^{-q}(\widetilde{w})]\}\mathbf{I}_{j}( \widetilde{w})|_{\widetilde{w}_{0}}&=-q\mathbf{H}\{\ln[\sigma_{m}(\widetilde{w})]\} \mathbf{I}_{j}(\widetilde{w})|_{\widetilde{w}_{0}}\\ &=-\frac{q}{\sigma_{m}(\widetilde{w})}\Bigg{[}\left\{\frac{\widetilde{q}^{2}}{ \widetilde{q}[\widetilde{w}]_{0}^{2}|\widetilde{w}]}\sigma_{m}(\widetilde{w} )\right\}_{\mathbf{I}_{j}=1,2,\ldots,2N_{\mathrm{R}}}\Bigg{]}\Big{|}_{\widetilde{w }=\widetilde{w}_{0}}\\ &=-\sum_{k=1}^{N}\frac{4(\widetilde{w}_{0}-\widetilde{c}_{m,k})(\widetilde{w} _{0}-\widetilde{c}_{m,k})^{T}\widetilde{c}_{m,k}(\widetilde{w}_{0})}{q\sigma^{2} \sigma_{m}(\widetilde{w})}-\frac{2}{\sigma^{2}}\mathbf{I}_{2N_{\mathrm{R}}}\prec 0 \end{array}  \tag{15}\]Note that (A5) is equivalent to an implicit function of \(\widetilde{c}_{m,1}\), \(\widetilde{c}_{m,2}\),..., \(\widetilde{c}_{m,N}\) and \(\widetilde{w}_{0}\) as below  
\[\mathcal{F}(\widetilde{c}_{m,1},\ldots,\widetilde{c}_{m,N},\widetilde{w}_{0}) =\left[\left\{f_{I}(\widetilde{c}_{m,1},\ldots,\widetilde{c}_{m,N}, \widetilde{w}_{0})\right\}\right]_{I=1,2,\ldots,2N_{\mathbb{R}}}\right]=0\] (A7)  
where  
\[ \begin{array}{l}f_{I}(\widetilde{c}_{m,1},\ldots,\widetilde{c}_{m,N}, \widetilde{w})\\ =\sum\limits_{k=1}^{N}\left([\widetilde{w}]_{j}-q[\widetilde{c}_{m,k}]_{j} \right)\exp\left(\frac{1}{q^{\sigma^{2}}}\sum\limits_{l=1}^{2N_{\mathbb{R}}} \left([\widetilde{w}]_{l}-q[\widetilde{c}_{m,k}]_{l}\right)^{2}-\frac{q+1}{ \sigma^{2}}\sum\limits_{l=1}^{2N_{\mathbb{R}}}\left[\widetilde{c}_{m,k}]_{l} \right]^{2}\right)\end{array} \] (A8)  
for \(i=1,2,\ldots,2N_{\mathbb{R}}\). Then the Jacobi matrix of \(\mathcal{F}(\widetilde{c}_{m,1},\ldots,\widetilde{c}_{m,N},\widetilde{w})\) is computed as below  
\[ \begin{array}{l}\mathbf{J}_{\mathcal{F}}|(\widetilde{c}_{m,1},\ldots, \widetilde{c}_{m,N},\widetilde{w})=\left[\left\{\frac{\partial}{\partial[ \widetilde{w}]_{l}}f_{I}(\widetilde{c}_{m,1},\ldots,\widetilde{c}_{m,N}, \widetilde{w})\right\}\right]_{I=1,2,\ldots,2N_{\mathbb{R}}}\right]\\ =\sigma_{m}(\widetilde{w})\left(\mathbf{I}_{2N_{\mathbb{R}}}+\frac{2}{q \sigma^{2}\widetilde{c}_{m}(\widetilde{w})}\sum\limits_{k=1}^{N}\left( \widetilde{w}-q\widetilde{c}_{m,k}\right)(\widetilde{w}-q\widetilde{c}_{m,k}) ^{\mathsf{T}}\sigma_{m,k}(\widetilde{w})\right)\end{array} \] (A9)  
Recalling (A6), \(\mathbf{J}_{\mathcal{F}}|(\widetilde{c}_{m,1},\ldots,\widetilde{c}_{m,N}, \widetilde{w})\) is a positive definite matrix at \(\widetilde{w}_{0}\), so it is invertible. Consequently, \(\widetilde{w}_{0}\) can, in principle, express in terms of \(\widetilde{c}_{m,1}\), \(\widetilde{c}_{m,2}\),..., \(\widetilde{c}_{m,N}\) by implicit function theorem. Namely, the maximum of \(\sigma_{m}\)\({}^{-q}(\widetilde{w})\) is achieved on the condition that \(\widetilde{w}_{0}\) satisfies (A5). Note that a complex number is zero when and only when both its real and imaginary parts are zero vectors, so we have,  
\[\mathbf{w}_{0}=\sum\limits_{k=1}^{N}q\rho_{m,k}c_{m,k}\triangleq\overset{-}{q} \overset{-}{c}_{m}\] (A10)  
where \(\rho_{m,k}=\sigma_{m,k}(\mathbf{w}_{0})/\sigma_{m}(\mathbf{w}_{0})\) is a positive real number over an open interval (0, 1) and satisfies \(\sum\
olimits_{k=1}^{N}\rho_{m,k}=1\). So \(\mathbf{w}_{0}\) and \(\overset{-}{c}_{m}\) are both weighted average vectors of \(c_{m,1}\), \(c_{m,2}\),..., and \(c_{m,N}\).