## 3 LiGME Model for Group Sparse Estimation  
### GME of Weighted \(\ell_{2,1}\)-Norm and Its Properties  
Although \(\ell_{2,1}\)-norm (or its weighted variants) acts as the favorable approach to approximate \(\ell_{2,0}\) in the literature of group sparse estimation, it has large bias and does not promote group sparsity as effective as \(\ell_{2,0}\). Since GME provides an approach to better approximate direct discrete measures (e.g., \(\ell_{0}\) for sparsity, matrix rank for low-rankness) than their convex envelopes, we propose to use it for designing group-sparsity pursuing regularizers. More generally, let us consider the GME of \(\|\cdot\|_{w,2,1}\) in (3). Clearly, \(\|\cdot\|_{w,2,1}\in\Gamma_{0}(\mathbb{R}^{n})\) is coercive, even symmetry and prox-friendly, whose proximity operator can be computed by  
\[\text{Prox}_{\gamma\|\cdot\|_{w,2,1}}:\ \mathbb{R}^{n}\to\mathbb{R}^{n}:\ \mathbf{x}\mapsto \left\{\left(1-\frac{\gamma w_{i}}{\max\{\|\mathbf{x}_{i}\|_{2},\gamma w_{i}\}} \right)\mathbf{x}_{i}\right\}_{i=1}^{g}, \tag{10}\]  
where \(\mathbf{x}=[\mathbf{x}_{1}^{\top},\mathbf{x}_{2}^{\top},\cdots,\mathbf{x}_{g}^{\top}]^{\top} \in\mathbb{R}^{n}\) is a signal with group structure, \(\mathbf{x}_{i}\in\mathbb{R}^{n_{i}}(i=1,2,\cdots,g)\) and \(\sum_{i=1}^{g}n_{i}=n\). Actually, the GME of \(\|\cdot\|_{w,2,1}\) with \(\mathbf{B}\in\mathbb{R}^{b\times n}\) (see (4)):  
\[\left(\|\cdot\|_{w,2,1}\right)_{\mathbf{B}}(\mathbf{x})=\sum_{i=1}^{g}w_{i}\|\mathbf{x}_{i} \|_{2}-\min_{\mathbf{v}\in\mathbb{R}^{n}}\left\{\sum_{i=1}^{g}w_{i}\|\mathbf{v}_{i}\|_ {2}+\frac{1}{2}\|\mathbf{B}(\mathbf{x}-\mathbf{v})\|_{2}^{2}\right\}, \tag{11}\]  
where \(\mathbf{v}_{i}\in\mathbb{R}^{n_{i}}(i=1,2,\cdots,g)\) and \(\mathbf{v}=[\mathbf{v}_{1}^{\top},\mathbf{v}_{2}^{\top},\cdots,\mathbf{v}_{g}^{\top}]^{\top} \in\mathbb{R}^{n}\), can serve as a parametric bridge between \(\|\cdot\|_{2,0}\) and \(\|\cdot\|_{w,2,1}\). **Proposition 2**. : _(GME of \(\|\cdot\|_{w,2,1}\) can bridge the gap between \(\|\cdot\|_{2,0}\) and \(\|\cdot\|_{w,2,1}\).) Let \(\mathbf{B}_{\gamma}:=\text{BlockDiag}(\frac{w_{1}}{\sqrt{\gamma}}\mathbf{I}_{n_{1}}, \frac{w_{2}}{\sqrt{\gamma}}\mathbf{I}_{n_{2}},\cdots,\frac{w_{g}}{\sqrt{\gamma}} \mathbf{I}_{n_{g}})\) for \(\gamma>0\), where \(w_{i}>0\) is the weight in (3) for \(i=1,\cdots,g\). Then, for any \(\mathbf{x}\in\mathbb{R}^{n}\),_  
\[\lim_{\gamma\downarrow 0}\frac{2}{\gamma}\left(\|\cdot\|_{w,2,1}\right)_{\mathbf{B}_{ \gamma}}(\mathbf{x})=\|\mathbf{x}\|_{2,0}. \tag{12}\]  
_Together with the fact that \(\left(\|\cdot\|_{w,2,1}\right)_{\mathbf{O}_{w\times n}}(\mathbf{x})=\|\mathbf{x}\|_{w,2,1}\) where \(\mathbf{O}_{n\times n}\in\mathbb{R}^{n\times n}\) is the zero matrix, the regularization term \(\frac{2}{\gamma}\left(\|\cdot\|_{w,2,1}\right)_{\mathbf{B}_{\gamma}}(\mathbf{x})\) can serve as a parametric bridge between \(\|\cdot\|_{2,0}\) and \(\|\cdot\|_{w,2,1}\). As a special case, the GME of \(\|\cdot\|_{2,1}\) can serve as a parametric bridge between \(\|\cdot\|_{2,0}\) and \(\|\cdot\|_{2,1}\)._Proof. : The regularization term \(\frac{2}{\gamma}(\|\cdot\|_{w,2,1})_{\mathbf{B}_{\gamma}}(\mathbf{x}):\mathbb{R}^{n }\rightarrow\mathbb{R}:[\mathbf{x}_{1}^{\top},\mathbf{x}_{2}^{\top},\cdots,\mathbf{x}_{g}^{ \top}]^{\top}\mapsto\sum_{l=1}^{g}\frac{2}{\gamma}\varphi_{i}(\mathbf{x}_{i})\), where  
\[\varphi_{i}(\mathbf{x}_{i}):=w_{i}\|\mathbf{x}_{i}\|_{2}-\min_{\mathbf{v}_{i}\in\mathbb{R} ^{n_{i}}}\left\{w_{i}\|\mathbf{v}_{i}\|_{2}+\frac{w_{i}^{2}}{2\gamma}\|\mathbf{x}_{i}- \mathbf{v}_{i}\|_{2}^{2}\right\}\]  
for \(i=1,\cdots,g\). By ([39], Example 24.20), we obtain  
\[\frac{2}{\gamma}\varphi_{i}(\mathbf{x}_{i})= \begin{cases}\frac{2w_{i}}{\gamma}\| \mathbf{x}_{i}\|_{2}-\frac{w_{i}^{2}}{\gamma^{2}}\|\mathbf{x}_{i}\|_{2}^{2},&\text{if }\|\mathbf{x}_{i}\|_{2}\leq \frac{\gamma}{w_{i}}\\ 1,&\text{otherwise}.\end{cases}  \tag{13}\]  
Then, we obtain  
\[\lim_{\gamma\downarrow 0}\frac{2}{\gamma}\varphi_{i}(\mathbf{x}_{i})= \begin{cases}0,& \text{if }\|\mathbf{x}_{i}\|_{2}=0,\\ 1,&\text{otherwise},\end{cases}