Algorithm 1: Partition-Based Clustered-Sparse Bayesian Learning Algorithm

Input: the received pilot signal ${\mathbf{y}}_{\mathbf{p}}$; the dictionary matrix ${\mathbf{U}}_{\mathbf{P}}$; the noise variance $\lambda$;

the length of discrete paths L; the maximum number of iterations $r_{max}$;

the maximum number of discrete paths in one cluster $L{C}_{max}$; the threshold for

prunning small hyperparameters ${\gamma }_{th}$; the threshold to stop the whole

algorithm $ϵ$.

Initialize: the list of path power ${\gamma }_{list}={\mathbf{1}}_L$; the number of clusters $C=1$;

the list of cluster structure $\mathbf{CLS}$ : ${\mathsf{\Gamma }}_1={\mathbf{I}}_L$, ${\mathbf{B}}_1={\mathbf{I}}_M$ and the delay range of

the 1st cluster $R_d=[0,L-1]$; the iteration counter $r=0$.

Channel Estimation:

1: ${\sum }_{\mathbf{0}}\leftarrow diag\left({\mathsf{\Gamma }}_1\otimes {\mathbf{B}}_1,{\mathsf{\Gamma }}_2\otimes {\mathbf{B}}_2,\dots ,{\mathsf{\Gamma }}_C\otimes {\mathbf{B}}_C\right)$.

2: ${\sum }_{\mathbf{h}}={\sum }_0-{\sum }_0{\mathbf{U}}_{\mathbf{P}}^H{\left(\lambda \mathbf{I}+{\mathbf{U}}_{\mathbf{P}}{\sum }_0{\mathbf{U}}_{\mathbf{P}}^H\right)}^{-1}{\mathbf{U}}_{\mathbf{P}}{\sum }_0$.

3: ${\mu }_{\mathbf{h}}\leftarrow {\lambda }^{-1}{\sum }_{\mathbf{h}}{\mathbf{U}}_{\mathbf{P}}^H{\mathbf{y}}_{\mathbf{p}}$.

Cluster Evolution:

4: for $d=1,2,\dots ,C$do

5:  for  $i=1,2,\dots ,L_d$ do

6:   ${\gamma }_{d,i}\leftarrow \frac{1}{M}Tr\left[{\mathbf{B}}_d^{-1}\left({\sum }_{\mathbf{h}}^{d,i}+{\mu }_{\mathbf{h}}^{d,i}{\left({\mu }_{\mathbf{h}}^{d,i}\right)}^H\right)\right]$, and update ${\gamma }_{list}$ with ${\gamma }_{d,i}$.

7:  end for

8:  $L_d^{\prime }\leftarrow min(L_d,L{C}_{max})$.

9:  ${\mathbf{B}}_d\leftarrow \frac{1}{L_d^{\prime }}{\displaystyle \sum _{i=1}^{L_d^{\prime }}}\frac{{\sum }_{\mathbf{h}}^{d,i}+{\mu }_{\mathbf{h}}^{d,i}{\left({\mu }_{\mathbf{h}}^{d,i}\right)}^H}{{\gamma }_{d,i}}$ using the $L_d^{\prime }$ most significant continuous paths.

10:  ${\beta }_d\leftarrow \frac{{\alpha }_1}{{\alpha }_0}$, where ${\alpha }_1$ and ${\alpha }_0$ can be obtained through ${\mathbf{B}}_d$.

11: end for

Cluster Partition:

12: ${\mathbf{p}}^{\left(0\right)}\leftarrow$ FindIndex$(0<{\gamma }_{list}<{\gamma }_{th})$.

13: if ${\mathbf{p}}^{\left(0\right)}$ is not empty then

14:  ${\gamma }_{list}^{\left(old\right)}\leftarrow {\gamma }_{list}$ and ${\gamma }_{list}\left[{\mathbf{p}}^{\left(0\right)}\right]\leftarrow \mathbf{0}$.

15:  $\mathbf{CLS}\leftarrow$ Split($\mathbf{CLS}$, ${\mathbf{p}}^{\left(0\right)}$), where $\mathbf{CLS}$ is splitted into C clusters

according to ${\mathbf{p}}^{\left(0\right)}$.

16:  ${\mathbf{p}}^{\left(1\right)}\leftarrow$ FindIndex$({\gamma }_{list}\ge {\gamma }_{th})$ and ${\mathbf{U}}_{\mathbf{P}}\leftarrow {\mathbf{U}}_{\mathbf{P}}[:,{\mathbf{p}}^{\left(1\right)}\otimes (1:M)]$.

17: end if

18: for $d=1,2,\dots ,C$do

19:  Update $R_d$, ${\mathsf{\Gamma }}_d$ and ${\mathbf{B}}_d$ in $\mathbf{CLS}$ according to ${\mathbf{p}}^{\left(0\right)}$, ${\gamma }_{list}$ and ${\beta }_d$, respectively.

20: end for

Check stopping conditions:

21: $r\leftarrow r+1$.

22: return the sub-part of $\mathbf{Channel}\mathbf{Estimation}$ until $r\ge r_{max}$ or

$\parallel {\gamma }_{list}-{\gamma }_{list}^{\left(old\right)}{\parallel }_2^2<ϵ$.

Output: the pruned channel ${\mu }_{\mathbf{h}}$ with the covariance matrix ${\sum }_{\mathbf{h}}$ and the

cluster list $\mathbf{CLS}$.