Wenzhong Yan^⋆, Di Jin^†, Zhidi Lin^⋆ and Feng Yin^⋆
闫文忠^⋆、金迪^†、林志迪^⋆ 和 尹峰^⋆

School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, China
香港中文大学科学与工程学院，中国深圳

^† Signal Processing Group, Technische Universitat Darmstadt, Darmstadt, Germany
^† 德国达姆施塔特工业大学信号处理小组

ABSTRACT 抽象

Graph neural networks (GNNs) are popular to use for classifying structured data in the context of machine learning. But surprisingly, they are rarely applied to regression problems. In this work, we adopt GNN for a classic but challenging nonlinear regression problem, namely the network localization. Our main findings are in order. First, GNN is potentially the best solution to large-scale network localization in terms of accuracy, robustness and computational time. Second, proper thresholding of the communication range is essential to its superior performance. Simulation results corroborate that the proposed GNN based method outperforms all state-of-theart benchmarks by far. Such inspiring results are theoretically justified in terms of data aggregation, non-line-of-sight (NLOS) noise removal and low-pass filtering effect, all affected by the threshold for neighbor selection. Code is available at https://github.com/ Yanzongzi/GNN-For-localization.
图形神经网络 （GNN） 通常用于在机器学习环境中对结构化数据进行分类。但令人惊讶的是，它们很少应用于回归问题。在这项工作中，我们采用 GNN 来解决一个经典但具有挑战性的非线性回归问题，即网络定位。我们的主要发现是有序的。首先，GNN 在准确性、鲁棒性和计算时间方面可能是大规模网络定位的最佳解决方案。其次，通信范围的适当阈值对于其卓越的性能至关重要。仿真结果证实，所提出的基于 GNN 的方法迄今为止优于所有最先进的基准。这些鼓舞人心的结果在理论上从数据聚合、非视距 （NLOS） 噪声去除和低通滤波效果来看是合理的，所有这些都受到邻居选择阈值的影响。代码可在 https://github.com/ Yanzongzi/GNN-For-localization 获取。

IndexTerms— Graph neural networks, large-scale, network localization, non-line-of-sight, thresholding.
IndexTerms— 图形神经网络、大规模、网络定位、非视距、阈值。

1.    INTRODUCTION 1. 引言

In recent years, graph neural networks (GNNs) have achieved many state-of-the-art results in various graph-related learning tasks such as node classification, link prediction and graph classification [1–4]. Comparing with the multi-layer perceptrons (MLPs), GNNs can exploit extra information from the edges. More concretely, each node aggregates information of its adjacent nodes instead of merely using its own [5,6]. Though being effective, GNN models are often used to deal with classification tasks. While regression problems are more challenging and constitute a larger body of practical signal processing applications. In this paper, we consider a classic yet challenging nonlinear regression problem, namely network localization [7].
近年来，图神经网络 （GNN） 在各种与图相关的学习任务中取得了许多最先进的成果，例如节点分类、链接预测和图分类 [1–4]。与多层感知器 （MLP） 相比，GNN 可以利用来自边缘的额外信息。更具体地说，每个节点都聚合了其相邻节点的信息，而不仅仅是使用自己的节点 [5,6]。尽管 GNN 模型很有效，但通常用于处理分类任务。而回归问题更具挑战性，并且构成了更大的实际信号处理应用程序。在本文中，我们考虑了一个经典但具有挑战性的非线性回归问题，即网络定位 [7]。

Network localization requires not only the measurements between agent nodes and anchor nodes, but also the measurements between the agent nodes themselves. In the past decades, a variety of canonical network localization methods have been developed, including: 1) maximum likelihood estimation based methods [7,8]; 2) least-squares estimation based methods [9]; 3) multi-dimensional scaling based methods [10]; 4) mathematical programming based methods [11, 12] and 5) Bayesian message passing based methods [9,13,14].
网络定位不仅需要 agent 节点和 anchor 节点之间的测量，还需要 agent 节点本身之间的测量。在过去的几十年里，已经开发了多种典型网络定位方法，包括：1） 基于最大似然估计的方法 [7,8];2） 基于最小二乘估计的方法 [9];3） 基于多维缩放的方法 [10];4） 基于数学规划的方法 [11， 12] 和 5） 基于贝叶斯消息传递的方法 [9,13,14]。

Ignoring the effect due to non-line-of-sight (NLOS) propagation will incur severe performance degradation of the aforementioned methods. For remedy, one could perform NLOS identification for each link and then either discard the NLOS measurements or suppress them robustly in the aforementioned methods [15]. However, accurate NLOS identification requires large-scale offline calibration and huge amount of manpower. The NLOS effect can also be dealt with from algorithmic aspect. By assuming that the NLOS noise follows a certain probability distribution, the maximum likelihood estimation based methods were developed in [16, 17]. However, model mismatch may cause severe performance degradation. In a recent work [18], network localization problem is formulated as a regularized optimization problem in which the NLOS-inducing sparsity of the ranging-bias parameters was additionally exploited. Unfortunately, all of these methods are computationally expensive for large-scale networks.
忽略非视距 （NLOS） 传播的影响将导致上述方法的性能严重下降。为了补救，可以对每个链路进行 NLOS 识别，然后丢弃 NLOS 测量值，或者以上述方法稳健地抑制它们 [15]。然而，准确的 NLOS 识别需要大规模的离线校准和大量的人力。NLOS 效应也可以从算法方面处理。通过假设 NLOS 噪声遵循一定的概率分布，在 [16， 17] 中开发了基于最大似然估计的方法。但是，模型不匹配可能会导致性能严重下降。在最近的一项工作 [18] 中，网络定位问题被表述为一个正则化优化问题，其中还利用了 NLOS 诱导的测距偏置参数稀疏性。不幸的是，对于大规模网络来说，所有这些方法的计算成本都很高。

In this paper, we propose a fresh GNN based network localization method that is able to achieve all desired merits at the same time. First, it provides extremely stable and highly accurate localization accuracy despite of severe NLOS propagations. Second, it does not require laborious offline calibration nor NLOS identification. Third, it is scalable to large-scale networks at an affordable computation cost. As far as we know, this is the first time that GNN has been applied to network localization.
在本文中，我们提出了一种全新的基于 GNN 的网络定位方法，该方法能够同时实现所有所需的优点。首先，尽管存在严重的 NLOS 传播，但它仍能提供极其稳定和高度准确的定位精度。其次，它不需要费力的离线校准，也不需要 NLOS 识别。第三，它可以以可承受的计算成本扩展到大规模网络。据我们所知，这是 GNN 第一次被应用于网络本地化。

The remainder of this paper is organized as follows. Our problem is first formulated in Section 2. In Section 3, we introduce a fresh GNN framework for network localization. Numerical results are provided in Section 4, followed by the theoretical performance justification in Section 5. Finally, we conclude the paper in Section 6.
本文的其余部分组织如下。我们的问题首先在第 2 节中提出。在第 3 节中，我们介绍了一个用于网络本地化的新 GNN 框架。第 4 节提供了数值结果，第 5 节提供了理论性能理由。最后，我们在第 6 节中总结了本文。

2.    PROBLEM FORMULATION
2. 问题制定

We consider a wireless network in two-dimensional (2-D) space, and extension to the 3-D case is straightforward. We let, without loss of generality, Sa = {1,2,...,N_l} be the set of indices of the anchors, whose positions p_i = [x_i,y_i]^⊤,i ∈ Sa are known, and Sb = {N_l + 1,N_l+2,...,N} be the set of indices of the agents, whose positions are unknown. A distance measurement made between any two nodes i and j is given by
我们考虑二维 （2-D） 空间中的无线网络，扩展到 3-D 情况很简单。在不失去一般性的情况下，我们设 Sa = {12,...,N_l} 为锚点的索引集，其位置 p_i = [x_{i，y i}]^⊤，i ∈ Sa 是已知的，而 Sb = {N_l + 1，N_l+2,...,N} 是其位置未知的代理的索引集。在任意两个节点 i 和 j 之间进行的距离测量由下式给出

                                         x_ij = d(p_i,p_j) + n_ij,                                     (1)
x_ij = d（p，p） + n_ij，                                     （1）

where d(p_i,p_j) := ∥p_i − p_j∥ is the Euclidean distance and n_ij is an additive measurement error due to line-of-sight (LOS) and NLOS propagation. A distance matrix, denoted by X ∈ R^N×N, is constructed by stacking the distance measurements, where x_ij is the (ij)-th entry of X. Notably, the distance matrix X is a “zerodiagonal” matrix, because x_ii = 0 for i = 1,2,...,N. Based on this distance matrix, our goal is to accurately locate the agents in a large-scale wireless network with satisfactory computation time.
其中 d（p，p） ：= ∥p_i − p∥ 是欧几里得距离，n_ij 是由于视距 （LOS） 和 NLOS 传播引起的加性测量误差。用 X ∈ R^N×N 表示的距离矩阵是通过堆叠距离测量值来构建的，其中 x_ij 是 X 的第 （ij） 个条目。值得注意的是，距离矩阵 X 是一个“零对角线”矩阵，因为 i = 12,...,N 时 x_ii = 0。基于这个距离矩阵，我们的目标是以令人满意的计算时间准确定位大规模无线网络中的智能体。

The above signal model well suits many realistic large-scale wireless networks. For instance, in 5G network, a large number of small base stations are densely deployed in each cell; Internet of Things (IoT) network, advocating interconnection of everything, comprises a huge number of connected smart devices and machines [19]. For large-scale networks, it is typical that only a small fraction of nodes know their locations precisely. To know all locations, otherwise, one requires either a lot of manpower to do offline calibration or to equip expensive and power-hungry GPS/BeiDou chips.
上述信号模型非常适合许多实际的大规模无线网络。例如，在 5G 网络中，每个小区中密集部署了大量小型基站;物联网 （IoT） 网络倡导万物互联，由大量互联的智能设备和机器组成 [19]。对于大规模网络，通常只有一小部分节点能够精确知道其位置。要知道所有位置，否则，要么需要大量人力进行离线校准，要么配备昂贵且耗电的 GPS/北斗芯片。

To locate a large number of agents, we propose a completely new learning paradigm, namely GNN based network localization, which is data-driven and relies on a graph-based machine learning model, to be specified in the next section.
为了定位大量智能体，我们提出了一种全新的学习范式，即基于 GNN 的网络定位，它是数据驱动的，依赖于基于图的机器学习模型，将在下一节中指定。

3.    NETWORK LOCALIZATION WITH GCN
3. 使用 GCN 进行网络本地化

Among different types of GNN models, graph convolutional networks (GCNs) constitute a representative class. In this section, we focus on formulating the network localization problem using GCNs. An undirected graph associated with a wireless network can be formally defined as G = (V,A), where V represents the vertex set of the nodes {v₁,v₂,...,v_N}, and A ∈ R^N×N is a symmetric adjacency matrix where a_ij denotes the weight of the edge between v_i and v_j. We set a_ij = 1 if there is no connection between v_i and v_j, otherwise a_ij = 0. The degree matrix D ∈ R^N×N := diag(d₁,d₂,...,d_N) is a diagonal matrix with.
在不同类型的 GNN 模型中，图卷积网络 （GCN） 构成了一个代表性类别。在本节中，我们重点介绍如何使用 GCN 构建网络定位问题。与无线网络关联的无向图可以正式定义为 G = （VA），其中 V 表示节点的顶点集 {v 1，v₂,...,v_N}，A ∈ R^N×N 是一个对称邻接矩阵，其中 _ij 表示 v_i 和 v_j.如果 v_i 和 v_j 之间没有连接，我们设置 _ij = 1，否则 _{a ij} = 0。度矩阵 D ∈ R^N×N ：= diag（d 1，d₂,...,d_N） 是一个对角矩阵。

In the original GCN, the edge a_ij can be regarded as a similarity coefficient between nodes i and j. In the context of network localization, two nodes are similar means that they are close to each other, i.e., d(p_i,p_j) being small. Accordingly, we introduce a Euclidean distance threshold, denoted by T_h, to determine whether there is an edge between two nodes or not. As will be explained in Section 5, this threshold is critical to the localization performance. By thresholding, a refined adjacency matrix A_Th ∈ R^N×N is constructed as follows:
在原始 GCN 中，边 _{a ij} 可以看作是节点 i 和 之间的相似系数。在网络定位的上下文中，两个节点相似意味着它们彼此靠近，即 d（p，p） 很小。因此，我们引入了一个欧几里得距离阈值，用 T_h 表示，以确定两个节点之间是否有边。正如第 5 节中将解释的那样，此阈值对于定位性能至关重要。通过阈值，构建一个精炼的邻接矩阵 A_Th ∈ R^N×N，如下所示：

                              [ATh]ij =  0,         if      xij > Th                                          (2)
[ATh]ij = 0，        如果      xij > Th                                          （2）

                                                  1,     otherwise 1，否则.

Consequently, the augmented adjacency matrix [1] is defined as A^˜ _Th := A_Th + I, where I is an identity matrix, and the associated degree matrix of A^˜ _Th is denoted by D^˜ _Th ∈ R^N×N
因此，增广邻接矩阵 [1] 定义为 A ^̃ _Th ：= A_Th + I，其中 I 是单位矩阵，而 A ^̃ _Th 的相关度矩阵用 D ^̃ _Th ∈ R 表示^{N× 号}.

Similarly, we construct a sparse distance matrix ^Xˆ = A_Th ⊙X, where ⊙ denotes the Hadamard product. Consequently, X^ˆ contains only distance measurements that are smaller than or equal to T_h
同样，我们构造了一个稀疏距离矩阵 ^Xˆ = A_Th ⊙X，其中 ⊙ 表示 Hadamard 积。因此，X^ˆ 仅包含小于或等于 T_h 的距离测量值.

In general, each layer of GCN carries out three actions: feature propagation, linear transformation and an element-wise nonlinear activation [20]. The main difference between GCN and the standard multi-layer perceptron (MLP) [21] lies in the feature propagation, which will be clarified in the following.
一般来说，GCN 的每一层都执行三个动作：特征传播、线性变换和元素级非线性激活 [20]。GCN 和标准多层感知器 （MLP） [21] 之间的主要区别在于特征传播，这将在下面进行澄清。

In the k-th graph convolution layer, we assume D_k is the number of neurons in the k-th layer, then the input and output node representations are denoted by the matrices H^(k−1) and H^(k) ∈ R^N×Dk, respectively. The initial node representations is H⁽⁰⁾ = X^ˆ . A K-layer GCN differs from a K-layer MLP in that the hidden representation of each node in GCN is averaged with its neighbors. More precisely, in GCN, the update process for all layers is obtained by performing the following matrix multiplication:
在第 k 个图卷积层中，我们假设 D_k 是第 k 层中的神经元数量，那么输入和输出节点表示分别用矩阵 H^（k−1） 和 H^（k） ∈ R^N×Dk 表示。初始节点表示为 H^（0） = X^ˆ 。K 层 GCN 与 K 层 MLP 的不同之处在于，GCN 中每个节点的隐藏表示是与其邻居的平均值。更准确地说，在 GCN 中，所有层的更新过程都是通过执行以下矩阵乘法获得的：

                                H¯ (k) ∈ RN×D^k−¹ ← Aˆ ThH(k−1),                           (3)
H ̄ （k） ∈ RN×D^k−¹ ← Aˆ ThH（k−1），                           （3）

where A^ˆ is the augmented normalized adjacency matrix [1] and H^{¯ (k)} is the hidden representation matrix in the k-th graph convolution layer. Intuitively, this step smoothes the hidden representations locally along the edges of the graph and ultimately encourages similar predictions among locally connected nodes.

Fig. 1: Diagram of GCN updating rule for one hidden layer.

After feature propagation, the remaining two steps of GCN, i.e., linear transformation and nonlinear activation, are identical to those of the standard MLP. The k-th layer contains a layer-specific trainable weight matrix W^(k) ∈ R^Dk−^1×Dk and a nonlinear activation function ϕ(·), such as ReLU(·) = max(0,·) [22]. The representation updating rule of the k-th layer, presented in Fig. 1, is given by

                                  H .                             (4)

It is noteworthy that the activation function ϕ(·) is applied to every element in the matrix.

Taking a 2-layer GCN as an example, the estimated positions, Rˆ = [pˆ₁,pˆ₂,...,pˆ_N]^⊤ ∈ R^N×2, are given by

                 R^ˆ  .            (5)

The weight matrices W⁽¹⁾ and W⁽²⁾ can be optimized by minimizing the mean-squared-error (MSE), L(W⁽¹⁾,W⁽²⁾) := ∥R_l −

R^ˆ_l∥²_F, where R_l = [p₁,p₂,...,p_Nl]^⊤ and R^ˆ_l = [pˆ₁,pˆ₂,...,pˆ_Nl]^⊤ are the true anchor positions and their estimates, respectively, and ∥· ∥F is the Frobenius norm of a matrix. This optimization problem is often solved via a gradient descent type method, such as stochastic gradient descent [23] or Adam [24].

4.    NUMERICAL RESULTS
4. 数值结果

In this section, the performance of the proposed GCN based method is evaluated in terms of localization accuracy, robustness against NLOS noise and computational time. For comparison purposes, we choose various well performed competitors, including an MLP based method, a neural tangent kernel (NTK) regression based method, the sparsity-inducing semi-definite programming (SDP) method [18], the expectation-conditional maximization (ECM) method [16], and the centralized least-square (LS) method [9]. Note that we choose MLP to demonstrate the performance improvement caused by adding the normalized adjacent matrix A^ˆ _Th in each layer. Additionally, we use NTK based regression to mimic ultra-wide MLP with random initialization based on the theorem that a sufficiently wide and randomly initialized MLP trained by gradient descent is equivalent to a kernel regression predictor with the NTK [26].

Implementation details of these methods are as follows. We use a 2-layer GCN with 2000 neurons in each hidden layer. We train GCN and MLP models for a maximum number of 200 epochs (full batch size) using Adam with a learning rate of 0.01. We initialize the weights using the routine described in [27] and normalize the input feature vectors along rows. Dropout rate [28] is set to 0.5 for all layers. The settings of NTK remain the same as described in [25]. The regularization parameter in SDP is set to λ = 0.05. For both the ECM and LS methods, the initial positions are randomly generated in the square area. All simulations are performed on a server computer equipped with 48 Inter Xeon E5-2650 2.2GHz CPUs and 8 NVIDIA TITAN Xp 12GB GPUs. In all experiments, we set the threshold T_h = 1.2 for GCN, MLP and NTK, and set T_h = 0.6 for other methods, which leads to similar averaged localization accuracy but requires much less computational time than using T_h = 1.2. Fairness in comparison is carefully maintained.

l

Fig. 2: The averaged loss (RMSE) versus the number of anchors under different noise conditions.

Details of the simulated localization scenarios are given below. We consider a 5m×5m unit square area. Each network consists of

500 randomly generated nodes in total. The number of anchors, N_l, varies from 20 to 160 for investigating its impact, and the rest are agents. The measurement error n_ij is generated according to n_ij = n^L_ij + b_ijn^N_ij. Here, the LOS noise n^L_ij is generated from a zero-mean Gaussian distribution, i.e., n^L_ij ∼ N(0,σ²), while the positive NLOS bias n^N_ij is generated from the uniform distribution, n^N_ij ∼ U[0,10] and b_ij generated from the Bernoulli distribution B(p_B) with p_B being the NLOS occurrence probability.

First, we assess the localization accuracy of all methods under different noise conditions. Here, the localization accuracy is measured in terms of the averaged test root-mean-squared-error (RMSE),

LR := ∥R_u −Rˆ_u∥F, where R_u = [p_Nl+1,p_Nl+2,...,p_N]^⊤ and

R^ˆ_u = [pˆ_Nl+1,pˆ_Nl+2,...,pˆ_N]^⊤. The results are summarized in Table 1. It is shown that among all considered methods, GCN provides the highest localization accuracy in almost all cases. In particular, when the NLOS probability, p_B, is high, GCN outperforms all competitors by far. Moreover, we test the localization performance of GCN for large networks with N = 1000, denoted by GCN₁₀₀₀ in Table 1. The results show that GCN performs even better with slightly increased computational time, as shown in Table 2. If we further increase N, the GCN based method can maintain its performance, but the other methods (SDP, ECM and LS) will all degrade severely in terms of localization accuracy and computational time.

h

Fig. 3: The averaged loss (RMSE) versus threshold under different noise conditions in GCN model. N_l = 50.

Next, we focus on two data-driven methods, GCN and MLP, which perform relatively well in Table 1. The localization accuracy is investigated by varying N_l from 20 to 160 with a stepsize of 20 under two different noise conditions. The results are depicted in Fig. 2. There are two main observations. First, GCN attains the lowest RMSE consistently for all N_l. Compared with MLP, the improvement in localization accuracy is particularly remarkable for GCN with small N_l. This result indicates that GCN can better exploit the distance information than MLP. When N_l increases, both GCN and MLP tend to approach a performance lower bound. Second, GCN performs similarly under both noise conditions, while MLP shows a clear performance degradation when p_B increases. This observation indicates that GCN is very robust against NLOS. Lastly, we want to mentioned that a fine-tuned MLP is often superior to NTK which corresponds to random initialized MLP, which performs surprisingly close to other benchmark methods as shown in Table 1.

In the third simulation, we focus on investigating the influence of the threshold, T_h, on the localization performance. Figure 3 depicts the RMSE of GCN versus the threshold, T_h, in three different noise scenarios. It is interesting to see that the RMSE curves show similar trend as the threshold changes. We characterize the localization performance obtained by using different thresholds. In Region I (T_h ∈ [0.2,0.8]), the RMSE is very large at the beginning and drops rapidly as T_h increases. The reason for such bad performance at the beginning is that when T_h is too small there will be no sufficient edges in the graph, incurring isolated nodes. In Regions II ∼ IV (T_h ∈ (0.8,2.8]), GCN shows stable performance. A closer inspection shows that the RMSE is relatively lower in Region II, rises slightly in Region III and decreases in Region IV to the same lowest level as in Region II. This observation can be explained as follows. When T_h ∈ (0.8,1.4], the good performance of GCN is due to the NLOS noise truncation effect of T_h, which will be explained in Section 5.1. For T_h ∈ (2.4,2.8], the adverse effect of large NLOS noise is compensated by the increased number of neighbors for each node. Lastly, the rapid increase of RMSE in Region V can be explained by the effect of extremely large NLOS noise and over-smoothing, which will be explained in Section 5.1 as well.

Table 2: A comparison of different methods in terms of computational time (in second) at (σ² = 0.1,p_B = 30%) and N_l = 50.

Another important requirement for real-world applications is fast response. Table 2 shows the practical computational time of different methods. It is shown that GCN, MLP and NTK are more computationally efficient than the other methods. Besides, the computational time of GCN₁₀₀₀ only slightly increases when we double the number of nodes in the network. Notably, GCN can process very large network, for instance N = 10000, in an affordable time, while the LS, ECM and SDP are all disabled in this case. All above results indicate that the proposed GCN based method is a prospective solution to large-scale network localization.

5.    PERFORMANCE REASONING
5. 性能推理

In this section, we aim to dig out the reasons behind the remarkable performance of the newly proposed GCN based method, corroborated by the results shown in Section 4. We pinpoint two major factors: one is the threshold T_h and the other is the augmented normalized adjacency matrix A^ˆ _Th. In the following, we analyze the two factors separately, although T_h determines A^ˆ _Th.

5.1. Effects of Thresholding

Thresholding plays two major roles: 1) truncating large noise and 2) avoiding over-smoothing.

Noise truncation. The operation, ^Xˆ = A_Th ⊙ X, in Section 3 implies that for each xˆ_ij = 0̸ , d(p_i,p_j) + n_ij ≤ T_h holds. Equivalently speaking, for each non-zero element in ^Xˆ , we have n_ij ≤ T_h − d(p_i,p_j), indicating that only noise in a small limited range will be included in X^ˆ . Specifically, due to the fact that n_ij with large value is usually caused by positive NLOS bias n^N_ij, each element x_ij, associated either with a large true distance or with a large noise, is neglected when the threshold T_h is set small. In other words, we retain the measurement if the two nodes are adjacent and only affected by a small or moderate measurement noise.

Avoiding over-smoothing. When the threshold is large enough, the corresponding graph will become fully connected and the adjacency matrix will be a matrix of all ones. In this case, according to Eq. (3), all entries of the hidden representation matrix H^{¯ (k)} are identical, meaning that the obtained hidden representation completely loses its distance information. Consequently, the predicted positions of all nodes will tend to converge to the same point. This phenomenon is known as over-smoothing. As an illustration, Region V in Fig. 3 confirms that GCN will suffer from over-smoothing when the threshold is set too large. Thus, we need to choose a proper threshold to prevent such an adversarial behavior.

5.2. Effects of A^ˆ _Th

To understand the superior localization performance of GCN compared with MLP, we analyze the effects of the augmented normalized adjacency matrix, A^ˆ _Th, from both spatial and spectral perspectives.

Aggregation and combination. To understand the spatial effect of A^ˆ _Th, we decompose A^ˆ _Th, cf. Eq. (3), into two parts: where h^¯(_i^k) and h⁽_i^k) are the i-th row vectors of hidden representation matrix, H^{¯ (k)}, and the input representation matrix, H^(k), in the k-th layer, respectively. Specifically, Eq. (6) contains two operations: aggregation and combination. Aggregation corresponds to the second term of Eq. (6), in which the neighboring features are captured by following the given graph structure. Then, the target node’s own information is combined with the aggregated information.

Fig. 4: Spectral components of different signals in dataset (σ² =

0.1,p_B = 0).

Own information

Aggregated information

Comparing with the training procedure of MLP, which solely uses the features of the labeled nodes (referred to as anchors here), GCN is a semi-supervised method in which the hidden representation of each labeled node is averaged for a carefully tailored local neighborhood including itself. Equivalently speaking, GCN trains a model by exploiting features of both labeled and unlabeled nodes, leading to superior localization performance.

Low-pass filtering. From the spectral perspective, the eigenvalues of the augmented normalized Laplacian L_Th = I − A^ˆ _Th, denoted by λ^˜, can be regarded as the "frequency" components [29,30]. Multiplying K augmented normalized adjacency matrices A^ˆin graph convolution layers is equivalent to passing a spectral “low-pass” filter g(λ^˜_i) := (1 − λ^˜_i)^K, where λ^˜_i,i = 1,2,... [20]. Figure 4 depicts the “frequency” components of the LOS noise and the true distance matrix before and after the filtering process. It can be seen that almost all information of the true distance matrix (before the filtering) is concentrated in the “low frequency” band, while both “low frequency” and “high frequency” components are present in the LOS noise before the filtering. Thus, A^ˆ _Th, acting as a “lowpass” filter, can partially remove the “high frequency” component of the LOS noise. This explains the improved localization performance of GCNs from the spectral perspective.

6.    CONCLUSION 6. 结论

In this paper, we have proposed a GCN based data-driven method for robust large-scale network localization in mixed LOS/NLOS environments. Numerical results have shown that the proposed method is able to achieve substantial improvements in terms of localization accuracy, robustness and computational time, in comparison with both MLP and various state-of-the-art benchmarks. Moreover, our detailed analyses found that thresholding the neighboring features is crucial to attaining superb localization performance. The proposed data-driven paradigm is believed to drive more efficient and robust methods for network localization and related ones in the future.