## 3 Methodology
### Simplify Belief Propagation  
According to the message update formula, the message \(m_{ut}\) transmitted by node \(U\) to node \(T\) is the accumulation of the product of the potential function between two nodes. There is an aggregation of messages transmitted by node \(U\) to neighboring nodes except \(T\) on the possible values of each node \(U\). Here, the information transmitted by the \(T\) node to the \(U\) node is removed to prevent the confidence of the \(T\) node from being used many times, resulting in overconfidence of the node. However, the approximation of LBP to BP shows that the repeated use of node confidence will not affect the expectation of node positioning, so here, we simplify the message from node \(U\) to node \(T\) as follows:  
\[m_{ut}^{i}(x_{t})=\int_{x_{u}}\psi_{tu}(x_{t},x_{u})B_{u}^{i-1}(x_{u})dx_{u} \tag{10}\]  
Because of the equivalence of the solution, we can make:  
\[m_{ut}^{i}(x_{t})=B_{u}^{i-1}(x_{u}) \tag{11}\]  
\[\Psi_{ut}(x_{t},x_{u})=\int_{x_{u}}\psi_{tu}(x_{t},x_{u})m_{ut}^{i}(x_{t})dx_{u} \tag{12}\]  
\[\tilde{B}_{t}^{i}(x_{t})=k\psi_{t}(x_{t})\prod_{u\in\tilde{G}_{t}}\Psi_{ut} \tag{13}\]  
In this way, we disassemble the SBP problem into three steps: The first step, the same as that shown in Formula (11), is to broadcast the belief of the node as the message of the node to all neighboring nodes. Secondly, according to the Formula (12), the potential function \(\Psi\) between the target node and the neighbor node is calculated based on the  
Figure 2: SBPVEM intra-node data flow. The node receives neighbors belief \(B_{i\in\mathcal{G}}\) and computes potential function \(\Psi\) between nodes under the constraints of distance measurement. The belief \(\tilde{B}\) can be obtained by multiplying all the neighboring \(\Psi\). \(\tilde{B}\) is complicated, by high-aggregation sampling and variation expectation–maximization, a GMM \(B\) is obtained to represent \(\tilde{B}\). belief of the source node and the constraint between the two nodes. In Step 3, based on Formula (13), the target node calculates the belief estimate of its own positioning \(\bar{B}\) based on the potential function. Assuming that the Gaussian mixture model can express the potential function of nodes, the Gaussian mixture model results from the superposition of multiple Gaussian distributions. \[G(\alpha,\mu,\Sigma)=\sum_{k=1}^{n}\alpha_{k}\mathcal{N}(\mu_{k},\Sigma_{k}) \tag{14}\]  
Among them, \(\mathcal{N}\) represents Gaussian distribution, and \(\mu_{k}\) and \(\Sigma_{k}\) are the parameters of the \(k\)-th Gaussian distribution. The formula of multi-dimensional Gaussian distribution is:  
\[\mathcal{N}(\mu_{k},\Sigma_{k})=\frac{1}{2\pi\sqrt{det(\Sigma_{k})}}\exp^{- \frac{1}{2}(x-\mu_{k})^{T}\Sigma_{k}^{-1}(x-\mu_{k})} \tag{15}\]  
The potential function of nodes is:  
\[B_{u}^{i}(x_{u})=G_{u}(\alpha,\mu,\Sigma) \tag{16}\]  
Formulas (14) and (16) are brought into (12), and there are:  
\[ \begin{split}&\Psi_{ut}(x_{t},x_{u})\\ &=\sum_{k=1}^{K}\alpha_{k}\int_{X_{u}}\psi_{tu}(x_{t},x_{u}) \mathcal{N}_{u}(\mu_{k},\Sigma_{k})dx_{u}\end{split}  \tag{17}\]  
According to Formulas (7) and (15), \(\psi_{tu}(x_{t},x_{u})\) can be regarded as one-dimensional Gaussian, and \(N_{u}(\mu_{k},\Sigma_{k})\) can be regarded as part of the integral that can be understood as the paired potential function \(\psi_{tu}\), the expectation under the probability \(\mathcal{N}_{u}(\mu_{k},\Sigma_{k})\). For the convenience of calculation, we can directly bring the paired potential function between nodes into the belief distribution of nodes. Let \(x_{s}\) be the equivalent variable:  
\[x_{s}=\frac{x_{u}-\mu}{|x_{u}-\mu|}(|x_{u}-\mu|-d) \tag{18}\]  
in which \(|x_{u}-\mu|-d\) considers the distance constraint between nodes and transforms the two-dimensional problem into one-dimensional, \(\frac{x_{u}-\mu}{|x_{u}-\mu|}\) keeps the directionality of the node position and restores the one-dimensional distance to a two-dimensional direction.