## 4 Discussion and Conclusions  
In this paper, we investigated the problem of denoising a contaminated Euclidean distance matrix (EDM) for high-accuracy connectivity-based localization based on the mathematical fact of an EDM, i.e., its low-rank property. Compared to conventional neural network (NN)-based algorithms with large-scale frameworks, we proposed two efficient algorithms, called denoising-autoencoder-aided EDM reconstruction (DAE-EDMR) and truncated DAE-EDMR (T-DAE-EDMR), which show a superior EDM denoising performance. Notably, the latter is designed within the NN framework, enabling it to achieve a robust performance even with a limited number of training datasets. Our contributions stem from the concept of inputting \(N\) (or \(k+2\) dominant) eigenvalues of the noisy EDM into the NN model, with the addition of marginal online complexity for eigenvalue decomposition (EVD) of the EDMs. Furthermore, T-DAE-EDMR reinforces the robustness of DAE-EDMR to variations in the number of nodes between the training and test phases. T-DAE-EDMR inputs the \(k+2\) dominant eigenvalues of the noisy EDM into the NN, extracted through \((k+2)\)-truncated EVD. This approach, as opposed to inputting \(N\) eigenvalues into DAE-EDMR, enhances the robustness to changing environments while  
Figure 4: Performance comparison according to the utilized matrices. (**a**) NMSE between ground-truth and denoised EDMs and (**b**) localization error in meters.  
reducing the required training dataset and off/online complexity. The proposed approach effectively leverages the linear algebraic properties of wireless localization. In particular, the use of eigenvalues to optimize the NN model for EDM denoising suggests that the volume of the convex hull formed by the distributed nodes can be interpreted as the crucial information. Our experimental results demonstrate that the proposed algorithms reduce the required training dataset's size to nearly one-tenth of its original size while achieving more than twice the effectiveness in EDM denoising. Given the suitability of the proposed methods for massive connectivity scenarios, our approach offers useful advantages for practical deployment. Furthermore, there is an opportunity to further enhance the EDM denoising performance by incorporating deeper mathematical insights, considering not only eigenvalues but also eigenvectors. Building on these strengths, future research will focus on extending our proposed schemes to localization and tracking techniques that can more robustly adapt to time-varying environments, making them even more applicable to large-scale implementations.  
Conceptualization: W.-H.L. and T.S. Methodology: W.-H.L. Software: W.-H.L. Validation: M.O. and U.C. Formal analysis: M.O. Investigation: U.C. Resources: W.-H.L. Data curation: W.-H.L. Writing--original draft preparation: W.-H.L. Writing--review and editing: W.-H.L., M.O., U.C., and T.S. Visualization: W.-H.L. Supervision: T.S. Project administration: T.S. Funding acquisition: W.-H.L. and T.S. All authors have read and agreed to the published version of the manuscript.  
This work was supported in part by an Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korean government (MSIT) (No. RS-2023-00217885, Development of integrated interference analysis technology for improving frequency utilization efficiency) and in part by the Soonchunhyang University Research Fund.  
Not applicable.  
The data supporting the reported results in this study can be shared upon reasonable request. Interested researchers should contact the corresponding author to discuss access. Please note that data sharing may be subject to restrictions due to privacy, ethical considerations, or intellectual property agreements with funders.  
Author [PERSON] was employed by the company Ericsson AB. The remaining authors declare that the re-search was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.