## Appendix B Parameter Tuning and Proximal Splitting Algorithm for LiGME Model  
**Proposition A1** ([24], Proposition 2). : _In (5), let \((\mathcal{X},\mathcal{Y},\mathcal{Z})=(\mathbb{R}^{n},\mathbb{R}^{m},\mathbb{ R}^{l})\), \((\mathbf{A},\mathfrak{L},\lambda)\in\mathbb{R}^{m\times n}\times\mathbb{R}^{l\times n }\times\mathbb{R}_{++}\), and \(\text{rank}(\mathfrak{L})=1\). Choose a nonsingular \(\mathfrak{L}\in\mathbb{R}^{n\times n}\) satisfying \([\mathbf{O}_{l\times(n-l)}\ \mathbf{I}_{l}]\mathfrak{L}=\mathfrak{L}\). Then \(\mathbf{B}_{\theta}:=\sqrt{\theta/\lambda}\mathbf{\Lambda}^{1/2}\mathbf{U}^{\top}\in \mathbb{R}^{l\times l}\), \(\theta\in[0,1]\), ensures \(J_{\mathbf{\Psi}_{\mathbf{B}_{\theta}\circ\mathfrak{L}}}\in\Gamma_{0}(\mathbb{R}^{n})\), where \([\tilde{\mathbf{D}}_{1}\,\tilde{\mathbf{D}}_{2}]:=\mathbf{A}\,(\mathfrak{L})^{-1}\) and \(\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{\top}:=\tilde{\mathbf{D}}_{2}^{\top}\tilde{\mathbf{D}}_{2}- \tilde{\mathbf{D}}_{2}^{\top}\tilde{\mathbf{D}}_{1}(\tilde{\mathbf{D}}_{1}^{\top}\tilde{ \mathbf{D}}_{1})^{\dagger}\tilde{\mathbf{D}}_{1}^{\top}\tilde{\mathbf{D}}_{2}\in\mathbb{R }^{l\times l}\) is an eigendecomposition._  
**Proposition A2** ([24], Theorem 1). : _Consider minimization of \(I_{\mathbf{\Psi}_{\mathbf{B}\circ\mathfrak{L}}}\) in (5) under the overall-convexity condition (6). Let a real Hilbert space \((\mathcal{H}:=\mathcal{X}\times\mathcal{Y}\times\mathcal{Z},\langle\cdot, \cdot\rangle_{\mathcal{H}},\|\cdot\|_{\mathcal{H}})\) be a product space and define an operator \(\mathcal{T}_{\text{LiGME}}:\mathcal{H}\rightarrow\mathcal{H}:(\mathbf{x},\mathbf{u}, \mathbf{v})\rightarrow(\mathfrak{L},\mathfrak{L},\mathbf{\eta})\) with parameters \((\iota,\tau)\in\mathbb{R}_{++}\times\mathbb{R}_{++}\), by_  
\[\mathfrak{L} :=\Big{[}\mathrm{Id}-\frac{1}{\sigma}(A^{*}A-\lambda\mathfrak{L} ^{*}\mathbf{B}^{*}\mathbf{B}\mathfrak{L})\Big{]}\mathbf{x}-\frac{\lambda}{\iota}\mathfrak{ L}^{*}\mathbf{B}^{*}\mathbf{B}\mathbf{u}-\frac{\lambda}{\iota}\mathfrak{L}^{*}\mathbf{v}+\frac{1}{ \iota}\mathbf{A}^{*}\mathbf{y},\] \[\mathfrak{L} :=\mathrm{Prox}_{\frac{1}{\tau}\mathbf{v}}\bigg{[}\frac{2\lambda}{ \tau}\mathbf{B}^{*}\mathbf{B}\mathfrak{L}\mathfrak{L}-\frac{\lambda}{\tau}\mathbf{B}^{*}\mathbf{B} \mathfrak{L}\mathbf{x}+\Big{(}\mathrm{Id}-\frac{\lambda}{\tau}\mathbf{B}^{*}\mathbf{B} \Big{)}\mathbf{u}\bigg{]},\] \[\mathbf{\eta} :=2\mathfrak{L}\mathfrak{L}-\mathfrak{L}\mathbf{x}+\mathbf{v}-\mathrm{Prox} _{\tau}(2\mathfrak{L}\mathfrak{L}-\mathfrak{L}\mathfrak{L}+\mathbf{v}).\]  
_Then the following holds:_  
1. \(\arg\min_{\mathbf{x}\in\mathcal{X}}\ J_{\mathbf{\Psi}_{\mathbf{B}\circ\mathfrak{L}}}(\mathbf{x})= \{\mathbf{x}^{\star}\in\mathcal{H}\mid(\mathbf{x}^{\star},\mathbf{u}^{\star},\mathbf{v}^{ \star})\in\mathrm{Fix}(\mathcal{T}_{\text{LiGME}})\}\)_, where_ \(\mathrm{Fix}(\mathcal{T}_{\text{LiGME}}):=\{(\mathbf{x},\mathbf{u},\mathbf{v})\in\mathcal{H} \mid\mathcal{T}_{\text{LiGME}}(\mathbf{x},\mathbf{u},\mathbf{v})=(\mathbf{x},\mathbf{u},\mathbf{v})\}\)_.__Choose \((\iota,\tau,\kappa)\in\mathbb{R}_{++}\times\mathbb{R}_{++}\times(1,\infty)\) satisfying_  
\[ \begin{split}&\iota\mathrm{Id}-\frac{\kappa}{2}\mathbf{A}^{*}\mathbf{A}- \lambda\mathfrak{L}^{*}\mathfrak{L}\succ\mathbf{O}_{\mathcal{X}},\\ &\tau\geq\Big{(}\frac{\kappa}{2}+\frac{2}{\kappa}\Big{)}\lambda \|\mathbf{B}\|_{\mathrm{op}}^{2},\end{split}  \tag{10}\]  
_where \(\|\cdot\|_{\mathrm{op}}\) is the operator norm. Then_  
\[\mathfrak{P}:= \begin{bmatrix}\iota\mathrm{Id}&-\lambda\mathfrak{L}^{*}\mathbf{B} ^{*}\mathbf{B}&-\lambda\mathfrak{L}^{*}\\ -\lambda\mathbf{B}^{*}\mathbf{B}\mathfrak{L}&\tau\mathrm{Id}&\mathbf{O}_{\mathcal{Z}}\\ -\lambda\mathfrak{L}&\mathbf{O}_{\mathcal{Z}}&\lambda\mathrm{Id}\end{bmatrix}  \succ\mathbf{O}_{\mathcal{H}} \tag{11}\]  
_and \(\mathcal{T}_{\mathrm{LiGME}}\) is \(\frac{\kappa}{2\kappa-1}\)-averaged nonexpansive in the Hilbert space \((\mathcal{H},\langle\cdot,\cdot\rangle_{\mathfrak{P}},\|\cdot\|_{\mathfrak{P }})\)._  
_Assume the condition (10) holds.