## 3 LiGME Model for Group Sparse Estimation
### GME of Weighted \(\ell_{2,1}\)-Norm and Its Properties
 \tag{14}\]  
and  
\[\lim_{\gamma\downarrow 0}\frac{2}{\gamma}(\|\cdot\|_{w,2,1})_{\mathbf{B}_{\gamma}}= \lim_{\gamma\downarrow 0}\sum_{i=1}^{g}\frac{2}{\gamma}q_{i}(\mathbf{x}_{i})=\|\mathbf{x} \|_{2,0}. \tag{15}\]  
Figure 1 illustrates simple examples of \(\|\mathbf{x}\|_{2,1}\) and \((\|\cdot\|_{2,1})_{\mathbf{B}}(\mathbf{x})\) when \(g=1\), \(n=2\) and \(\mathbf{B}=\mathbf{I}_{2}\). As we can see, \((\|\cdot\|_{2,1})_{\mathbf{I}_{2}}(\mathbf{x})\) can approximate \(\|\mathbf{x}\|_{2,0}\) better than \(\|\mathbf{x}\|_{2,1}\). Of course, as reviewed in Section 2.1, we can minimize \(J_{(\|\cdot\|_{w,2,1})_{\mathbf{B}}\circ\text{ld}}\) (see (5)) with the algorithm in (A3) in Proposition A2, under the overall-convexity condition \(\mathbf{A}^{\top}\mathbf{A}-\lambda\mathbf{B}^{\top}\mathbf{B}\succeq\mathbf{O}_{n\times n}\). In the following, we consider the GME of non-separable weighted \(\ell_{2,1}\)-norm \(\|\mathbf{W}\cdot\|_{2,1}\), where \(\mathbf{W}\in\mathbb{R}^{l\times n}\) is not necessarily a diagonal matrix. This is because in some applications, such as classification problems [11; 36; 37] stated in Section 2.2, and also heterogeneous feature selection [40], weights are introduced inside groups as well (i.e., the weight of every entry can be different) to improve the estimation accuracy. The GME of \(\|\mathbf{W}\cdot\|_{2,1}\) with \(\tilde{\mathbf{B}}\in\mathbb{R}^{b\times n}\) is well-defined (The lack of coercivity requires slight modification from min to inf.) as  
\[(\|\mathbf{W}\cdot\|_{2,1})_{\tilde{\mathbf{B}}}(\mathbf{x})=\|\mathbf{W}\mathbf{x}\|_{2,1}-\inf_{ \mathbf{v}\in\mathbb{R}^{n}}\biggl{\{}\|\mathbf{W}\mathbf{v}\|_{2,1}+\frac{1}{2}\|\tilde{ \mathbf{B}}(\mathbf{x}-\mathbf{v})\|_{2}^{2}\biggr{\}}, \tag{16}\]  
Figure 1: Simple examples of two group sparse regularizers (one group case): (a) The \(\ell_{2,1}\) regularizer; (b) The regularizer \((\|\cdot\|_{2,1})_{\mathbf{I}_{n}}\). and therefore we can formulate  
\[\underset{\mathbf{x}\in\mathbb{R}^{n}}{\text{minimize}}\ J_{(\|\mathbf{W}\cdot\|_{2,1})_{ \mathbf{B}:\text{id}}}=\frac{1}{2}\|\mathbf{y}-\mathbf{A}\mathbf{x}\|_{2}^{2}+\lambda(\|\mathbf{W} \cdot\|_{2,1})_{\tilde{\mathbf{B}}}(\mathbf{x}). \tag{17}\]  
However, we should remark that \(\|\mathbf{W}\cdot\|_{2,1}\in\Gamma_{0}(\mathbb{R}^{n})\) is even symmetric but not necessarily coercive or prox-friendly (As found in ([39], Proposition 24.14), it is known that for \(\Psi\in\Gamma_{0}(\mathcal{Z})\) and \(\mathfrak{L}\in\mathcal{B}(\mathcal{X},\mathcal{Z})\) satisfying \(\mathfrak{L}\mathfrak{L}^{*}=\mu\text{d}\) with some \(\mu\in\mathbb{R}_{++}\), we have \(\text{Prox}_{\Psi\circ\mathfrak{L}}(\mathbf{x})=\mathbf{x}+\mu^{-1}\mathfrak{L}^{*}( \text{Prox}_{\Psi\pi}(\mathfrak{L}\mathfrak{x})-\mathfrak{L}\mathbf{x})\) for \(\mathbf{x}\in\mathcal{X}\). In such a special case, if \(\Psi\) is prox-friendly, \(\Psi\circ\mathfrak{L}\) is also prox-friendly. However, for general \(\mathfrak{L}\in\mathcal{B}(\mathcal{X},\mathcal{Z})\) not necessarily satisfying such standard conditions, we have to discuss the prox-friendliness of \(\Psi\circ\mathfrak{L}\) case by case.). Fortunately, by Proposition 3 below, if \(\text{rank}(\mathbf{W})=l\) and \(\tilde{\mathbf{B}}\) can be expressed as \(\tilde{\mathbf{B}}=\mathbf{B}\mathbf{W}\) for some \(\mathbf{B}\in\mathbb{R}^{b\times l}\), we can show the useful relation  
\[(\|\mathbf{W}\cdot\|_{2,1})_{\tilde{\mathbf{B}}}(\mathbf{x})=(\|\cdot\|_{2,1})_{\mathbf{B}} \circ\mathbf{W}(\mathbf{x}), \tag{18}\]  
which implies that the GME \((\|\mathbf{W}\cdot\|_{2,1})_{\tilde{\mathbf{B}}}\) of \(\|\mathbf{W}\cdot\|_{2,1}\) can be handled as the LiGME \((\|\cdot\|_{2,1})_{\mathbf{B}}\circ\mathbf{W}\) of \(\|\cdot\|_{2,1}\). Proposition 3: _For \(\Psi\in\Gamma_{0}(\mathcal{Z})\) which is coercive and \(\mathbf{B}\in\mathcal{B}(\mathcal{Z},\tilde{\mathcal{Z}})\), assume \(\mathfrak{L}\in\mathcal{B}(\mathcal{X},\mathcal{Z})\) has full row-rank.