Table 1 PARALIND decomposition algorithm

Step 1:

Initialization

Initialize matrices ${\mathbf{J}}^{\left(0\right)}$,${\mathbf{A}}_{\theta }^{\left(0\right)}$,${\mathbf{F}}_{\varphi }^{\left(0\right)}$ and ${\mathbf{S}}^{\left(0\right)}$,$k=0$

Step 2:

$k\text{=}k\text{+}1$

Step 3:

Update J based on (21), given ${\mathbf{S}}^{\left(k\right)},{\mathbf{F}}_{\varphi }^{\left(k\right)}$and ${\mathbf{A}}_{\theta }^{\left(k\right)}$:

$\begin{array}{l}{\mathbf{j}}^{(k+1)}={({\mathbf{S}}^{\left(k\right)}\otimes {\mathbf{F}}_{\varphi }^{\left(k\right)}\odot {\mathbf{A}}_{\theta }^{\left(k\right)})}^{†}\overline{\mathbf{x}}\\ {\mathbf{J}}^{(k+1)}=unvec\left({\mathbf{j}}^{(k+1)}\right)\end{array}$

Step 4:

Update S based on (22), given ${\mathbf{F}}_{\varphi }^{\left(k\right)},{\mathbf{A}}_{\theta }^{\left(k\right)}$ and ${\mathbf{J}}^{(k+1)}$:

${\mathbf{S}}^{(k+1)}={\left(({\mathbf{F}}_{\varphi }^{\left(k\right)}\odot {\mathbf{A}}_{\theta }^{\left(k\right)}){\left({\mathbf{J}}^{(k+1)}\right)}^T\right)}^{†}\overline{\mathbf{X}}$

Step 5:

Update ${\mathbf{F}}_{\varphi }$ based on (23), given ${\mathbf{A}}_{\theta }^{\left(k\right)},{\mathbf{S}}^{(k+1)}$ and ${\mathbf{J}}^{(k+1)}$:

${\mathbf{F}}_{\varphi }^{(k+1)}={\left({\mathbf{A}}_{\theta }^{\left(k\right)}\odot \left({\mathbf{S}}^{(k+1)}{\mathbf{J}}^{(k+1)}\right)\right)}^{†}\overline{\mathbf{Z}}$

Step 6:

Update ${\mathbf{A}}_{\theta }$ based on (24), given ${\mathbf{S}}^{(k+1)},{\mathbf{J}}^{(k+1)}$ and ${\mathbf{F}}_{\varphi }^{(k+1)}$:

${\mathbf{A}}_{\theta }^{(k+1)}={\left(\left({\mathbf{S}}^{(k+1)}{\mathbf{J}}^{(k+1)}\right)\odot {\mathbf{F}}_{\varphi }^{(k+1)}\right)}^{†}\overline{\mathbf{Y}}$

Step 7:

Calculate the fitting residual value:

${\varepsilon }^{(k+1)}=\left|\right|({\mathbf{F}}_{\varphi }^{(k+1)}\odot {\mathbf{A}}_{\theta }^{(k+1)}){\left({\mathbf{S}}^{(k+1)}{\mathbf{J}}^{(k+1)}\right)}^T-({\mathbf{F}}_{\varphi }^{\left(k\right)}\odot {\mathbf{A}}_{\theta }^{\left(k\right)}){\left({\mathbf{S}}^{\left(k\right)}{\mathbf{J}}^{\left(k\right)}\right)}^T|{|}_F$

repeat step 2-7 until $|{\varepsilon }^{(k+1)}-{\varepsilon }^{\left(k\right)}|<1e^{-8}$.