## 2 Preliminaries  
### Review of Linearly Involved Generalized-Moreau-Enhanced (LiGME) Model  
We first give a brief review of linearly involved generalized-Moreau-enhanced (LiGME) models, which is closely related to our method. Although the convex function \(\ell_{1}\)-norm (or nuclear norm) is the most frequently adopted regularizer for sparsity (or low-rank) pursuing problems, it tends to yield underestimation for high-amplitude value (or large singular value) [26; 27]. The convexity-preserving nonconvex regularizers have been widely explored in [28; 29; 30; 31; 32; 33], which promote sparsity (or low-rank) more effectively than convex regularizers without losing the overall convexity. Among them, the generalized minimax concave (GMC) function in [31] does not rely on certain strong assumptions in the least squares term and has great potential for dealing with nonconvex variations of \(\|\cdot\|_{1}\). Motivated by GMC function, the LiGME model [24] provides a general framework for constructing linearly involved nonconvex regularizers for sparsity (or low-rank) regularized linear least squares while maintaining the overall convexity of the cost function. Let \((\mathcal{X},\langle\cdot\cdot\rangle_{\mathcal{X}}\|\cdot\|_{\mathcal{X}})\), \((\mathcal{Y},\langle\cdot\cdot\rangle_{\mathcal{Y}},\|\cdot\|_{\mathcal{Y}})\), \((\mathcal{Z},\langle\cdot\cdot\rangle_{\mathcal{Z}},\|\cdot\|_{\mathcal{Z}})\), and \((\mathcal{\tilde{Z}},\langle\cdot\rangle_{\mathcal{\tilde{Z}}},\|\cdot\|_{ \mathcal{\tilde{Z}}})\) be finite-dimensional real Hilbert spaces. Let a function \(\Psi\in\Gamma_{0}(\mathcal{Z})\) be coercive with \(\mathrm{dom}\Psi=\mathcal{Z}\). Here \(\Gamma_{0}(\mathcal{Z})\) is the set of proper (i.e., \(\mathrm{dom}\Psi:=\{z\in\mathcal{Z}|\Psi(z)<\infty\}\
eq\varnothing\)) lower semicontinuous (i.e., \(\mathrm{lev}_{\leq\alpha}\Psi:=\{z\in\mathcal{Z}|\Psi(z)\leq a\}\) is closed for \(\forall a\in\mathbb{R}\)) convex function (i.e., \(\Psi(\theta z_{1}+(1-\theta)z_{2})\leq\theta\Psi(z_{1})+(1-\theta)\Psi(z_{2})\)) for \(\forall z_{1},z_{2}\in\mathrm{dom}\Psi,0\leq\theta\leq 1\)) from \(\mathcal{Z}\) to \((-\infty,\infty]\); a function \(\Psi\in\Gamma_{0}(\mathcal{Z})\) is called coercive if \(\|z\|_{2}\to\infty\Rightarrow\Psi(z)\to\infty\). For \(\Psi\in\Gamma_{0}(\mathcal{Z})\), the proximity operator of \(\Psi\) is defined by \(\mathrm{Prox}_{\Psi}:\mathcal{Z}\to\mathcal{Z}:z\mapsto\arg\min_{v\in\mathcal{Z }}\ \Psi(v)+\frac{1}{2}\|v-z\|_{\mathcal{Z}}^{2}\). The generalized Moreau enhancement (GME) of \(\Psi\) with \(B\in\mathcal{B}(\mathcal{Z},\mathcal{\tilde{Z}})\) is defined as  
\[\Psi_{B}(\cdot):=\Psi(\cdot)-\min_{v\in\mathcal{Z}}\biggl{[}\Psi(v)+\frac{1}{ 2}\|B(\cdot-v)\|_{\mathcal{Z}}^{2}\biggr{]}, \tag{4}\]  
where \(B\) is a tuning matrix for the enhancement. Then the LiGME model defined as the minimization of  
\[J_{\Psi_{B}\circ\mathcal{Z}}:\ \mathcal{X}\to\mathbb{R}:\ x\mapsto\frac{1}{2}\|y- A\|_{\mathcal{Y}}^{2}+\lambda\Psi_{B}\circ\mathfrak{L}(x), \tag{5}\]  
where \((A,\mathfrak{L},\lambda)\in\mathcal{B}(\mathcal{X},\mathcal{Y})\times \mathcal{B}(\mathcal{X},\mathcal{Z})\times\mathbb{R}_{+}\). Please note that GMC [31] can be seen as a special case of (5) with \(\Psi=\|\cdot\|_{1}\) and \(\mathfrak{L}=\mathrm{Id}\), where \(\mathrm{Id}\) is the identity operator. Model (5) can also be seen as an extension of [32; 33]. Although the GME function \(\Psi_{B}\) in (4) is not convex in general for \(B\
eq O_{\mathcal{B}(\mathcal{Z},\mathcal{\tilde{Z}})}\), where \(O_{\mathcal{B}(\mathcal{Z},\mathcal{\tilde{Z}})}\in\mathcal{B}(\mathcal{Z}, \mathcal{\tilde{Z}})\) is the zero operator, the overall convexity of the cost function (5) can be achieved with \(B\) designed to satisfy the following convexity condition. **Proposition 1** ([24], Proposition 1).