## 2 Related Work  
The most well-known multi-hop distributed localization algorithm is the DV-hop proposed by [PERSON] and [PERSON] [10]. In the DV-hop method, the distances between the locating nodes to the anchor nodes can be represented by the product of network average hop distance and the hop-counts between them, then to obtain the location information via trilateration. For example, the average hop distance of anchor \(i\) can be calculated by Equation (1):  
Figure 1: Different distribution of nodes (**a**) Isotropic network; (**b**) Anisotropic network. \[c_{i}=\frac{\sum_{i\
eq j}d_{ij}}{\sum_{i\
eq j}h_{ij}} \tag{1}\]  
where \(d_{ij}\) is the physical distance between anchor node \(i\) and anchor node \(j\), and \(h_{ij}\) is the corresponding shortest hop-count. The DV-Hop algorithm is simple and easy to implement. The whole localization process does not rely on additional hardware devices. However, its adaptability to the environment is poor, therefore the applications are limited. By virtue of machine learning methods, modeling and designing the localization mechanism has become a hot topic in recent years [11; 12]. Based on the correspondence between the distributed characteristics and the measured information of the known nodes, a mapping model can be learned. Then, this model is used to estimate the position of the unknown nodes. Compared with the previous work, such model can effectively find out the intrinsic network topology, the relevance of observations and other information hidden in large-scale data, which then effectively improve the localization performance. [PERSON] et al. [9] proposed a Proximity Distance Map (PDM) localization algorithm using Truncated Singular Value Decomposition (TSVD) technology to model hop-counts and physical distance [13]. The PDM method first records the physical distance and the measured distance between the known nodes with the matrix respectively. Secondly, it transforms the two matrices linearly through the TSVD method to obtain an optimal linear transformation model. Thirdly, it takes the shortest hop-count from unknown nodes to anchor nodes into the model. Finally, the physical distance of unknown nodes to anchor nodes is estimated. The TSVD is essentially a linear multivariate rule by which the estimated distance obtained is the weighted sum of the estimated values of the known nodes associated with the unknown nodes, and thus the estimated values are closer to the true values. In addition, TSVD drops the smaller singular values. To a certain extent, this operation reduces the impact of noise in the conversion process, so as to enhance the stability of the algorithm. However, the PDM method ignores the problem of scale conversion between the hop-counts matrix and the physical distances matrix, which can cause data loss. In addition, it does not offer a choice for TSVD clip coefficients. Accordingly, an excessive coefficient may easily cause useful information lost, while a too small clip coefficient would retain too much high-frequency noise. Recently, inspired by the PDM method, [PERSON] and others proposed multi-hop range-free localization methods based on SVM [14; 15], namely, Localization through Support Vector Regression (LSVR) and Localization through Multi-dimensional Support Vector Regression (LMSVR) [7; 8]. These two methods suggest that there is a nonlinear relationship between the hop-counts and the physical distances. They use a regression based on the kernel technology SVM to predict the distance between nodes. However, according to [PERSON]'s razor principle, the more complex the model, the higher the probability of over-fitting [16]. Therefore, the localization method based on SVM algorithm has the over-fitting phenomenon with the increase of the number of anchor nodes, which resulted in the reduction of the localization accuracy. In addition, SVM-based methods are prone to excessive computational complexity. The [PERSON] team from Hong Kong Polytechnic University put forward another strategy.