###### Abstract
Regarding the online complexity, the proposed method requires eigenvalue decomposition (EVD) of the EDM as a pre-processing step for the NN operations. In addition, we propose a technique called truncated DAE-EDMR, i.e., _T-DAE-EDMR_, to enhance the robustness of DAE-EDMR to diverse environments conditions, such as changes in the number of nodes, even after NN optimization. In other words, T-DAE-EDMR offers the flexibility to be utilized in scenarios where the number of nodes varies between the training and test phases. The T-DAE-EDMR scheme involves feeding \(k+2\) eigenvalues from the noisy EDM to the NN, which are extracted through (\(k+2\))-truncated EVD. In contrast, DAE-EDMR inputs \(N\) eigenvalues, making T-DAE-EDMR more versatile  
Figure 1: An illustration of an example of the proposed DAE-EDMR (Additionally, truncated DAE-EDMR (T-DAE-EDMR) is presented as a relaxed version of DAE-EDMR. This additional work is carried out to make the optimized NN model more efficient by using the dominant \(k+2\) eigenvalues as the input data. Details can be found in Section 2.3). in accommodating different environments and simultaneously reducing the online complexity. Hence, T-DAE-EDMR is envisioned to demonstrate its superior effectiveness in environments characterized by a large number of nodes or frequent topology changes, such as vehicle-to-everything (V2X) systems. To summarize, the main contributions of the proposed algorithms to denoising the noisy EDMs and enhancing the localization accuracy are four-fold:  
* **Minimizing the reconstruction errors for the EDM through the mix-up of mathematical evidence and NNs**: We reveal the potential to reconstruct the EDM according to the low-rank property of the ground-truth EDM and the low-dimensional representations of the NN operation, which is robust to various noise models in distance measurements. * **Reducing the size of the training dataset and NN model**: We develop an efficient NN framework requiring a small-sized training dataset and NN model. This is based on the novel inputting and outputting for the NN model, which consist of the eigenvalues of the noisy EDM and the ground-truth EDM, respectively. * **Assisting the existing connectivity-based localization algorithms as a pre-processor**: We combine our proposed scheme with the connectivity-based localization technique to validate its utility in various environments. It is verified that these joint frameworks show a superior performance compared to that of the conventional approaches. This achievement is highly remarkable, as we only need to extract the eigenvalues of an EDM, requiring marginal online complexity. * **Making the NN model robust to the dynamics of wireless networks**: By additionally presenting a modified model of our proposed model, we introduce an NN that can be robust to the variability in wireless networks, e.g., the number of nodes in the test phase is changed after NN optimization. The remainder of this paper is organized as follows: Section 2 presents the system model, the problem design, and the method for the proposed algorithms. In Section 3, we provide numerical results to demonstrate the superiority of our proposed algorithms compared to other schemes even given the aspects of various environments. Finally, Section 4 presents the concluding remarks.