Algorithm 1: HVB-DCS Algorithm

Input: A set of measurement vectors $\{{\mathbf{y}}_1,{\mathbf{y}}_2,\cdots ,{\mathbf{y}}_K\}$ and corresponding measurement matrices

$\{{\mathbf{A}}_1,{\mathbf{A}}_2,\cdots ,{\mathbf{A}}_K\},k=1,\cdots ,K$.

Output: The reconstructed signal ${\widehat{\mathbf{\rho }}}_k={\mathbf{\mu }}_c+{\mathbf{\mu }}_k$, $k=1,\cdots ,K$.

Initialize the hyperparameters. Set the initial values of the variables (a, b, $\{c_k,k=1,\cdots ,K\}$,

$\{d_k,k=1,\cdots ,K\}$, e, f) as ${10}^{-6}$.

Compute the variational distribution for the common component.

Compute ${\mathsf{\Sigma }}_c={\left({\displaystyle \sum _{k=1}^K}\langle \beta \rangle {\mathbf{A}}_k^T{\mathbf{A}}_k+{\mathsf{\Lambda }}_{\langle {\mathbf{\alpha }}_c\rangle }\right)}^{-1}$, and ${\mathbf{\mu }}_c=\langle \beta \rangle {\mathsf{\Sigma }}_c{\displaystyle \sum _{k=1}^K}{\mathbf{A}}_k^T({\mathbf{y}}_k-{\mathbf{A}}_k{\mathbf{\mu }}_k).$

Compute the variational distribution for the prior of the common component.

Update $q\left({\alpha }_c^n\right)$, compute

${\displaystyle \tilde{a}=a+\frac{1}{2},{\tilde{b}}_n=b+\frac{1}{2}〈{\left(z_c^n\right)}^2〉.}$

Compute the variational distributions for the innovation components.

Compute ${\mathsf{\Sigma }}_k={\left(\langle \beta \rangle {\mathbf{A}}_k^T{\mathbf{A}}_k+{\mathsf{\Lambda }}_{\langle {\mathbf{\alpha }}_k\rangle }\right)}^{-1}$, and ${\mathbf{\mu }}_k=\langle \beta \rangle {\mathsf{\Sigma }}_k{\mathbf{A}}_k^T\left({\mathbf{y}}_k-{\mathbf{A}}_k{\mathbf{\mu }}_c\right)$.

Compute the variational distributions for the prior of the innovation components.

Update $q\left({\alpha }_k^n\right)$, compute

${\displaystyle {\tilde{c}}_k=c_k+\frac{1}{2},{\tilde{d}}_k^n=d_k+\frac{1}{2}〈{\left(z_k^n\right)}^2〉.}$

Compute the variational distribution for the prior of noise vector.

Update $q\left(\beta \right)$, compute

${\displaystyle \tilde{e}=e+\frac{KM}{2},}$

${\displaystyle \tilde{f}=f+\frac{1}{2}\left(\sum _{k=1}^K{〈{∥{\mathbf{y}}_k-{\mathbf{A}}_k({\mathbf{z}}_c+{\mathbf{z}}_k)∥}_2^2〉}_{q\left({\mathbf{z}}_c\right)q\left({\mathbf{z}}_k\right)}\right).}$

Iterate steps 2 , 3 , 4 , 5 and 6 until convergence occurs to hyperparameters.

Output ${\widehat{\mathbf{\rho }}}_k={\mathbf{\mu }}_c+{\mathbf{\mu }}_k$ for $k=1,\cdots ,K$.