Algorithm 1 NetSCCA: Network constrained sparse common component analysis.

1: Compute jaccard similarity: $\mathit{W}$.

2: For q target genes, compute the regulator effect matrices as ${\mathit{R}}_{\ell }$ for $\ell =1,\dots ,q$ and

$\mathit{G}={\sum }_{\ell =1}^q{\mathit{R}}_{\ell }^T{\mathit{R}}_{\ell }$.

3: For the square root of $\mathit{G}$ (i.e., ${\mathit{Q}}^T\mathit{Q}=\mathit{G}$), compute sparse common loadings of q

regulator effect matrices ${\mathit{R}}_{\ell },\ell =1,\dots ,q$.

3.1: Start $\mathit{A}$ at $\mathit{V}=[V_1,V_2,\dots ,V_K]$, which is the loading matrix from ordinary PCA of $\mathit{Q}$.

3.2: Given a fixed $\mathit{A}=[{\mathit{a}}_1,{\mathit{a}}_2,\dots ,{\mathit{a}}_K]$, solving the following problem,

$\begin{array}{c}{\widehat{\mathit{\theta }}}_k\underset{{\theta }_k}{\arg \min }\{\parallel {\mathit{z}}_k-\mathit{Q}{\mathit{\theta }}_k{\parallel }^2\}+{\lambda }_1{\parallel {\mathit{\theta }}_k\parallel }_1+{\lambda }_2\sum _{j<s}{({\theta }_{j,k}-{\theta }_{s,k})}^2{W}_{j,s}k=1,2,\dots ,K,\end{array}$

where ${\mathit{z}}_k=\mathit{Q}{\mathit{a}}_k$. Update $\widehat{\mathbf{\Theta }}=[{\widehat{\mathit{\theta }}}_1,{\widehat{\mathit{\theta }}}_2,\dots ,{\widehat{\mathit{\theta }}}_K]$.

3.3: For a fixed $\widehat{\mathbf{\Theta }}$, perform the singular value decomposition of ${\mathit{Q}}^T\mathit{Q}\widehat{\mathbf{\Theta }}=\mathit{U}\Gamma {\mathit{V}}^T$ and

update $\widehat{\mathit{A}}=\mathit{U}{\mathit{V}}^T$ (see Zou et al. [25]).

3.4: Repeat Steps 3.2–3.3, until convergence.

4: Sparse common loading is given by $\frac{{\widehat{\mathit{\theta }}}_k}{\parallel {\widehat{\mathit{\theta }}}_k\parallel }$ for $k=1,\dots ,K$.