Algorithm 1: GaBP Algorithm

Step 0 Initialization: Set a convergence threshold $ϵ$, $P_{ki}=0$ and ${\theta }_{ki}=0,\forall k\in N(i)$. Compute $P_{ii}={\mathcal{A}}_{ii}$ and ${\theta }_{ii}=b_i/{\mathcal{A}}_{ii}$. Step 1 Iteration: Propagate the messages $P_{ki}$ and ${\theta }_{ki},\forall k\in N(i)$. Compute $P_{ij}=-{\mathcal{A}}_{ij}^2/(P_{ii}+{\displaystyle \sum _{k\in N(i)\backslash j}}{P}_{ki}),{\theta }_{ij}=(P_{ii}{\theta }_{ii}+{\displaystyle \sum _{k\in N(i)\backslash j}}{P}_{ki}{\theta }_{ki})/{\mathcal{A}}_{ij}$. Step 2 Convergence check: If the message $P_{ij}$ and ${\theta }_{ij}$ do not converge, return to Step 1, else, go to Step 3. Step 3 Inference: Compute the marginal means ${\theta }_i=(P_{ii}{\theta }_{ii}+{\displaystyle \sum _{k\in N(i)}}{P}_{ki}{\theta }_{ki})/((P_{ii}+{\displaystyle \sum _{k\in N(i)}}{P}_{ki}))$. Step 4 Output: Output the solution $w_i={\theta }_i$.