Algorithm 1: EM algorithm for GBFA
P ← {j : xj is Poisson}.
Initialize m, W, ρ, and θ.
$$\begin{array}{c}\mathbf{repeat}\hfill \\ \mid \begin{array}{c}\mathbf{for}\underset{¯}{l=1,\dots ,L}\mathbf{do}∕\star \text{E}-\text{step for}\gamma \star ∕\hfill \\ \mid \begin{array}{c}{\alpha }_{\mathit{jl}}\leftarrow {\alpha }_{\mathit{jl}}+s_l(-\delta +\eta \sum \begin{array}{c}\hfill \\ k\hfill \end{array}{G}_{\mathit{jk}}{\theta }_{\mathit{kl}}+\text{log}({\lambda }_1∕{\lambda }_0)-\mid w_{\mathit{jl}}\mid ({\lambda }_1-{\lambda }_0)-{\alpha }_{\mathit{jl}})\text{for all}j\text{where}{s}_l\text{is determined by the}\hfill \\ \text{backtracking line search}.\hfill \\ {\theta }_{\mathit{jl}}\leftarrow 1∕(1+e^{-{\alpha }_{\mathit{jl}}});{\lambda }_{\mathit{jl}}\leftarrow (1-{\theta }_{\mathit{jl}}){\lambda }_0+{\theta }_{\mathit{jl}}{\lambda }_1\text{for all}j.\hfill \end{array}\phantom{\mid }\hfill \\ \mathbf{end}\hfill \\ \mathbf{if}\underset{¯}{\mathcal{P}=\varnothing }\mathbf{then}\text{update}{\mu }_{z,i}\text{and}{\Sigma }_{z,i}\text{as in (9) for}1\le i\le n.∕\star \text{E}-\text{step for}Z\star ∕\hfill \\ \mathbf{else}\hfill \\ \mid \mathbf{for}\underset{¯}{i=1,\dots ,n}\mathbf{do}\text{update}{\mu }_{z,i}\text{and}{\Sigma }_{z,i}\text{by the gradient ascent method with the backtracking line search}.\hfill \\ \mathbf{end}\hfill \\ \mathbf{for}\underset{¯}{j=1,\dots ,p}\mathbf{do}\text{update}{\phi }_{\mathit{ji}},{\rho }_{\mathit{ji}},\text{and}{\rho }_j\text{as in Section III-B2}.∕\star \text{E}-\text{step for}\rho \star ∕\hfill \\ \mathbf{for}\underset{¯}{j=1,\dots ,p}\mathbf{do}∕\star \text{M}-\text{step for}W\star ∕\hfill \\ \mid \begin{array}{c}\mathbf{if}\underset{¯}{j\notin \mathcal{P}}\mathbf{then}{\tilde{\mathbf{w}}}_j\leftarrow {\text{argmin}}_{\mathbf{w}}(\frac{1}{2}{\mathbf{w}}^T{A}_j\mathbf{w}-{\mathbf{b}}_j^T\mathbf{w}+\sum \begin{array}{c}\hfill \\ l\hfill \end{array}{\lambda }_{\mathit{jl}}\mid w_l\mid \text{where}{A}_j\text{and}{\mathbf{b}}_j\text{are given by (10)}.\hfill \\ \mathbf{else}\text{update}{\tilde{\mathbf{w}}}_j\text{by the proximal gradient ascent method}.\hfill \end{array}\phantom{\mid }\hfill \\ \mathbf{end}\hfill \end{array}\phantom{\mid }\hfill \\ \mathbf{until}\underset{¯}{\text{convergence}}.\hfill \end{array}$$