## 1 Introduction  
Positioning is essential in multi-sensor networks, vehicle ad hoc networks [1], underwater unmanned clusters [2], military environment research [3], forest emergency rescue [4], and other tasks that need environmental awareness. Cooperative localization uses measurements between unlocalized target nodes to localize all network nodes when anchor nodes are not sufficient or accessible.  
Collaborative localization based on filtering [5] and optimization [6] requires that the nodes have a good initial state. The optimization result is unreliable when the target node does not have a good prior. However, nodes cannot provide good and reliable prior under divergent conditions and some extreme conditions. Refs. [7, 8] use belief propagation (BP) methods to deal with the problems of no initial position of target nodes, few anchor nodes, and sparse links between nodes. The BP method is an accurate reasoning method applied to an acyclic probability graph, and the loopy belief propagation (LBP) method approximates the BP algorithm on an acyclic graph. [PERSON]. [9] proposed a SPAWN method for location on the basis of LBP. In order to solve the positioning probability in continuous space, the BP algorithm needs to discretize the value space by the grid. The high probability interval may be smaller than the feasible region, which wastes many computational resources of the algorithm. [PERSON]. [10] proposed the Nonparametric Belief Propagation (NBP) method, which uses weighted particles to represent the irregular distribution of node positions. However, communication overhead is the critical problem in the implementation, which mainly comes from the representation of information exchanged between neighboring nodes. The particle-based method requires a higher communication overhead between nodes.  
In order to reduce the communication overhead, [PERSON]. [11] combines the idea of a sigma point to fit the fitted parameters, which uses a Gaussian to represent the information between nodes. Refs. [12, 13] show that when BP based on parameter fitting is applied to the positioning problem, it can only deal with the situation that the positioning center of nodes is Gaussian distribution. Ref. [14] considers the multimodal situation of node position but confuses Gaussian distribution and mixed Gaussian distribution. Select a Gaussian to fit the mixed Gaussian. Ref. [15] proposed for the first time to use the VMP algorithm as a generalization of BP on a factor graph and to use a variational method to fit the probability of nodes, which can fit the non-Gaussian situation. However, the mean-field hypothesis destroys the association between multi-dimensional states. Ref. [16] applies the VMP method to the positioning problem. However, only single-mode and dual-mode positioning cases are considered. Ignore the constraint range of other nodes when the node can only communicate with a few anchor nodes.  
This paper proposes the SBPVEM method for the distributed cooperative localization of sparse anchor targets in the algorithm. A simplified belief propagation algorithm is proposed as the overall reasoning framework by modeling the cooperative localization problem as a graph model. The high-aggregation sampling and variation expectation-maximization algorithm is applied to sample and fit the complicated distribution. In practice, our work solves the self-localization problem of distributed nodes under the condition of a minimal number of anchor nodes, makes cooperative localization with sparse anchor nodes in wireless sensor networks a reality, and lowers the cost in large-scale applications.  
The Section 1 of this paper details the subject's background and associated studies. Section 2 models cooperative localization problems as a graph model and describes belief propagation as necessary background knowledge. The proposed SBPVEM approach is detailed in depth in the Section 3. The Section 4 includes some experiments to show our algorithm's effectiveness. Discussion is presented in the Section 5.