## 4 Bayesian Learning-Based Clustered-Sparse Channel Estimation
### Cluster Distribution Learning  
The hyperparameters \(\mathbf{\theta}\triangleq[\mathbf{\gamma}_{1},\mathbf{\gamma}_{2},\cdots,\mathbf{\gamma}_{C},\beta_{1},\beta_{2},\cdots,\beta_{C}]\) can be estimated by maximizing the marginal likelihood function \(p(\mathbf{y}_{\mathbf{p}};\mathbf{\theta})\). This is equivalent to minimizing \(-log\ p(\mathbf{y}_{\mathbf{P}};\mathbf{\theta})\), giving the effective cost function:  
\[\mathcal{L}(\mathbf{\theta})=\mathbf{y}_{\mathbf{P}}^{H}\mathbf{\Sigma}_{\mathbf{y}_{ \mathbf{p}}}^{-1}(\mathbf{\theta})\mathbf{y}_{\mathbf{p}}+log|\mathbf{\Sigma}_{ \mathbf{y}}(\mathbf{\theta})|, \tag{22}\]  
where \(\mathbf{\Sigma}_{\mathbf{y}_{\mathbf{p}}}(\mathbf{\theta})=\lambda\mathbf{I}+\mathbf{ U}_{\mathbf{P}}\mathbf{\Sigma}_{0}(\mathbf{\theta})\mathbf{U}_{\mathbf{P}}^{H}\). The above problem cannot be solved in closed form, and the expectation maximization (EM) method [41] can be employed to solve it iteratively. For unknown values of hyperparameters governing the prior density, \(\mathbf{h}\) is considered as the nuisance variable and \(\mathbf{\theta}\) is estimated by maximizing:  
\[Q(\mathbf{\theta}|\mathbf{\theta}^{(old)})=E_{\mathbf{h}|\mathbf{y}_{\mathbf{p}};\mathbf{ \theta}^{(old)}}[log\ p(\mathbf{y}_{\mathbf{p}},\mathbf{h};\mathbf{\theta})]=E_{ \mathbf{h}|\mathbf{y}_{\mathbf{p}};\mathbf{\theta}^{(old)}}[log\ p(\mathbf{y}_{ \mathbf{p}}|\mathbf{h};\lambda)]+E_{\mathbf{h}|\mathbf{y}_{\mathbf{p}};\mathbf{ \theta}^{(old)}}[log\ p(\mathbf{h};\mathbf{\theta})], \tag{23}\]  
where \(\mathbf{\theta}  
the first term unrelated to \(\mathbf{\theta}\), the Q Function (23) can be simplified to:  
\[Q(\beta_{1},\beta_{2},\dots,\beta_{C} \gamma_{1},\gamma_{2},\dots,\gamma_{C})=E_{\mathbf{h}|\mathbf{y}_{ \mathbf{P}},\mathbf{\theta}^{(a|d)}}\{log\,p[\mathbf{h}_{\mathbf{;}}\,\mathbf{B}_{ 1},\,\mathbf{B}_{2},\cdots,\mathbf{B}_{C},\mathbf{\Gamma}_{1},\mathbf{\Gamma}_{2}, \cdots,\mathbf{\Gamma}_{C}]\}\] \[\propto -\frac{1}{2}\sum_{d=1}^{C}\left[log\,(|\mathbf{\Gamma}_{d}|^{M}| \mathbf{B}_{d}|^{L_{d}})+\mathbf{h}_{d}^{H}(\mathbf{\Gamma}_{d}^{-1}\otimes \mathbf{B}_{d}^{-1})\mathbf{h}_{d}\right]\] \[\propto -\frac{1}{2}\sum_{d=1}^{C}\left\{Mlog\,(|\mathbf{\Gamma}_{d}|)+L_{d} log\,(|\mathbf{B}_{d}|)+Tr\Big{[}\Big{(}\mathbf{\Gamma}_{d}^{-1}\otimes\mathbf{B}_{d} ^{-1}\Big{)}\Big{(}\mathbf{\Sigma}_{\mathbf{h}}^{d}+\mathbf{\mu}_{\mathbf{h}}^{d}(\bm {\mu}_{\mathbf{h}}^{d})^{H}\Big{)}\Big{]}\right\}\]  
where \(\mathbf{h}_{d}\in\mathbb{C}^{ML_{d}\times 1}\), \(\mathbf{\mu}_{\mathbf{h}}^{d}\in\mathbb{C}^{ML_{d}\times 1}\) and \(\mathbf{\Sigma}_{\mathbf{h}}^{d}\in\mathbb{C}^{ML_{d}\times ML_{d}}\) denote the corresponding \(d\)th cluster in \(\mathbf{h}\in\mathbb{C}^{ML\times 1}\), \(\mathbf{\mu}_{\mathbf{h}}\in\mathbb{C}^{ML\times 1}\) and \(\mathbf{\Sigma}_{\mathbf{h}}\in\mathbb{C}^{ML\times ML}\), respectively. \(\mathbf{\mu}_{\mathbf{h}}\) and \(\mathbf{\Sigma}_{\mathbf{h}}\) are evaluated according to (19) and (20), given the estimated hyperparameters \(\mathbf{\theta}^{(a|d)}\) and the noise variance \(\lambda\). As shown in (14) and (16), \(\mathbf{\Gamma}_{d}\) and \(\mathbf{B}_{d}\) are determined by the hyperparameters \(\gamma_{d}\) and \(\beta_{d}\), respectively. The partial derivative of (24) with respect to \(\gamma_{d,j}\) (\(d=1,2,\dots,C;i=1,2\dots,L_{d}\)) is given by  
\[\frac{\partial Q}{\partial\gamma_{d,i}}=-\frac{M}{2\gamma_{d,i}}+\frac{1}{2 \gamma_{d,i}^{2}}Tr\bigg{[}\mathbf{B}_{d}^{-1}\bigg{(}\mathbf{\Sigma}_{\mathbf{h} }^{d,i}+\mathbf{\mu}_{\mathbf{h}}^{d,i}\Big{(}\mathbf{\mu}_{\mathbf{h}}^{d,i}\Big{)}^ {H}\bigg{)}\bigg{]}, \tag{25}\]  
where \(\mathbf{\mu}_{\mathbf{h}}^{d,i}\in\mathbb{C}^{M\times 1}\) denotes the corresponding \(i\)th path in \(\mathbf{\mu}_{\mathbf{h}}^{d}\), and \(\mathbf{\Sigma}_{\mathbf{h}}^{d,i}\in\mathbb{C}^{M\times M}\) is the corresponding principal diagonal block for the \(i\)th path in \(\mathbf{\Sigma}_{\mathbf{h}}^{d}\).