## 3 Proposed Fusion Algorithm
### Abundance Estimation via ADMM  
Using ADMM and Kronecker operators, we can integrate two Frobenius-norm terms into an \(\ell_{2}\)-norm term by vectorizing the variable \(\mathbf{S}\). In (11), the bottleneck lies in the heavy complexities of \(\mathbf{B}_{1}\) and \(\mathbf{H}\). By splitting the primal variables into several separable components via equality constraints, we rewrite (11) as  
\[\min_{\mathbf{s},\mathbf{u},\mathbf{v},\mathbf{x}} \ \frac{1}{2}\|\mathbf{B}_{1}\mathbf{s}-\mathbf{y}\|_{2}^{2}+ \lambda_{h}\|\mathbf{v}\|_{1}+\lambda_{s}\|\mathbf{x}\|_{1}+I_{+}(\mathbf{x})\] (13) s.t. \[\ \mathbf{s}=\mathbf{u},\] \[\ \mathbf{v}=\mathbf{H}\mathbf{u},\] \[\ \mathbf{s}=\mathbf{x},\]  
where \(\mathbf{B}_{1}\triangleq[(\mathbf{G}^{T}\otimes\mathbf{A}^{k})^{T}\), \((\mathbf{I}_{L}\otimes\mathbf{FA}^{k})^{T}]^{T}\in\mathbb{R}^{(ML_{h}+LM_{m}) \times NL}\); \(\mathbf{y}\triangleq[\mathbf{vec}(\mathbf{Y}_{h})^{T}\), \(\mathbf{vec}(\mathbf{Y}_{m})^{T}]^{T}\in\mathbb{R}^{ML_{h}+LM_{m}}\); and \(I_{+}(\mathbf{x})\) is a indicator function, defined as  
\[I_{+}(\mathbf{x})\triangleq \begin{cases}0,&\text{if}\ \mathbf{x}\succeq\mathbf{0}_{MN},\\ \infty,&\text{otherwise}.\end{cases} \]  
In addition, the equation constraint \(\mathbf{s}=\mathbf{u}\) is introduced to separate the \(\mathbf{s}\) and \(\mathbf{v}\) using the variable \(\mathbf{u}\) for reducing the complexity of closed-form solutions. Then, we can obtain the augmented Lagrangian of (13) by  
\[\mathcal{L}(\mathbf{s},\mathbf{u},\mathbf{v},\mathbf{x},\mathbf{h }_{i})= \ \frac{1}{2}\|\mathbf{B}_{1}\mathbf{s}-\mathbf{y}\|_{2}^{2}+ \lambda_{h}\|\mathbf{v}\|_{1}+\lambda_{s}\|\mathbf{x}\|_{1}+I_{+}(\mathbf{x} )+\mathbf{h_{1}}^{T}(\mathbf{s}-\mathbf{u})+\frac{\eta}{2}\|\mathbf{s}-\mathbf{ u}\|_{2}^{2}\] \[\ +\mathbf{h_{2}}^{T}(\mathbf{v}-\mathbf{H}\mathbf{u})+\frac{\eta}{2 }\|\mathbf{v}-\mathbf{H}\mathbf{u}\|_{2}^{2}+\mathbf{h_{3}}^{T}(\mathbf{s}- \mathbf{x})+\frac{\eta}{2}\|\mathbf{s}-\mathbf{x}\|_{2}^{2}, \tag{14}\]where \(\mathbf{h}_{i}(i=1,\cdots,3)\) are called dual variables, and \(\eta\) is the weight. Via ADMM, these primal and dual variables can be iteratively updated in step with provable convergence [46] by  
\[\mathbf{s}^{j+1} \in\arg\min_{\mathbf{s}\in\mathbb{R}^{N\times L}}\mathcal{L}( \mathbf{s},\mathbf{u}^{j},\mathbf{x}^{j},\mathbf{h}_{1}{}^{j},\mathbf{h}_{3}{}^ {j}), \tag{15a}\] \[\mathbf{u}^{j+1} \in\arg\min_{\mathbf{u}\in\mathbb{R}^{N\times L}}\mathcal{L}( \mathbf{s}^{j+1},\mathbf{u},\mathbf{v}^{j},\mathbf{h}_{1}{}^{j},\mathbf{h}_{2}{ }^{j}),\] (15b) \[\mathbf{v}^{j+1} \in\arg\min_{\mathbf{v}\in\mathbb{R}^{2N\times L}(\sqrt{L}-1)} \mathcal{L}(\mathbf{u}^{j+1},\mathbf{v},\mathbf{h}_{2}{}^{j}),\] (15c) \[\mathbf{x}^{j+1} \in\arg\min_{\mathbf{x}\in\mathbb{R}^{N\times L}_{+}}\mathcal{L} (\mathbf{s}^{j+1},\mathbf{x},\mathbf{h}_{3}{}^{j}),\] (15d) \[\mathbf{h}_{1}{}^{j+1} =\mathbf{h}_{1}{}^{j}+\eta(\mathbf{s}^{j+1}-\mathbf{u}^{j+1}),\] (15e) \[\mathbf{h}_{2}{}^{j+1} =\mathbf{h}_{2}{}^{j}+\eta(\mathbf{v}^{j+1}-\mathbf{H}\mathbf{u} ^{j+1}),\] (15f) \[\mathbf{h}_{3}{}^{j+1} =\mathbf{h}_{3}{}^{j}+\eta(\mathbf{s}^{j+1}-\mathbf{x}^{j+1}), \tag{15g}\]  
where \(j\) denotes the iteration number of inner loop.