O-SBL Algorithm:

${\left\{{\mathsf{\Theta }}^{(i)}\right\}}_{i=1}^{N_{\mathrm{collect}}}=\mathbf{O}-\mathbf{SBL}(Y,A,{\mathsf{\Theta }}_0,N_{\mathrm{burn}-\mathrm{in}},N_{\mathrm{collect}})$

For $\mathrm{Iter}=1\mathrm{to}{N}_{\mathrm{burn}-\mathrm{in}}+N_{\mathrm{collect}}$

% Support-learning vector component

For $p=1\mathrm{to}P$

${\tilde{y}}_{mn}^{-p}=y_{mn}-{\sum }_{l\ne p}^P{a}_{ml}{s}_l{x}_{ln}$, $\forall m=1\mathrm{to}M,\forall n=1\mathrm{to}N$

$q_0=1-{\gamma }_p$

$q_1={\gamma }_p{e}^{-\frac{\epsilon }{2}({∥{\mathbf{a}}_p∥}_2^2{\sum }_{n=1}^N{x}_{pn}^2-2{\mathbf{a}}_p^T{\sum }_{n=1}^N{x}_{pn}{\tilde{\mathbf{y}}}_n^{-p})}$

$(s_p|-)\sim \mathrm{Bernoulli}(\frac{q_1}{q_0+q_1})$

% Solution-value matrix component

For $l=1\mathrm{to}P$

${\sigma }_x={(\tau +\epsilon s_l^2{∥{\mathbf{a}}_l∥}_2^2)}^{-1}$

$\overline{\mathit{\mu }}=\epsilon s_l{\sigma }_x{\mathbf{a}}_l$

${\tilde{\mathbf{y}}}_n^{-l}={\mathbf{y}}_n-A(\mathbf{s}\circ {\mathbf{x}}_n)+s_l{x}_{ln}{\mathbf{a}}_l,\forall n=1\mathrm{to}N$

$(x_{ln}|-)\sim \mathcal{N}({\overline{\mathit{\mu }}}^T{\tilde{\mathbf{y}}}_n^{-l},{\sigma }_x),\forall n=1\mathrm{to}N$

End For{l}

$({\gamma }_p|-)\sim \mathrm{Beta}({\alpha }_0+s_p,{\beta }_0+1-s_p)$

End For{p}

$(\tau |-)\sim \mathrm{Gamma}(a_0+\frac{NP}{2},b_0+\frac{1}{2}{∥X∥}_F^2)$

$(\epsilon |-)\sim \mathrm{Gamma}({\theta }_0+\frac{MN}{2},{\theta }_1+\frac{1}{2}{∥Y-A(\mathbf{s}\circ X)∥}_F^2)$

${\mathsf{\Theta }}^{(\mathrm{Iter}-N_{\mathrm{burn}-\mathrm{in}})}\leftarrow \mathsf{\Theta }$, $\forall \mathrm{Iter}>N_{\mathrm{burn}-\mathrm{in}}$

End For{Iter}