###### Abstract
Errors in the EDM may lead to inaccurate and imprecise location estimations, which will result in a less reliable and robust connectivity-based localization process. Hence, it is critical to address the EDM contamination issue to ensure accurate and efficient localization in large-scale IoT networks. To mitigate the effects of noisy measurements on the EDM, denoising techniques have been extensively utilized, and they aim to denoise a noisy EDM by resolving properly designed optimization problems such as semi-definite relaxation [16; 17] and low-rank tensor completion [18]. However, they rely on solving complex optimization problems, which may limit their efficiency and effectiveness. Developments in artificial intelligence (AI) methods, particularly neural network (NN)-based denoising methods such as [19; 20; 21; 22], have become promising in leveraging statistical inference as a novel and potentially more robust alternative for dealing with noisy measurements in EDM-based localization. The existing NN-based EDM denoising techniques require computations that scale with the square of the total number of nodes since both the input and output of the NN framework are based on pairwise distances. This is due to the combinatorial nature of measuring and generating the distances between node pairs. As the number of nodes increases, the computational complexity grows quadratically, leading to the need for a large training dataset size and extensive NN models. In addition, the existing NN-based denoising methods may not be suitable for IoT networks due to their resource limitations. Although major research efforts have focused on big data analysis and deep neural networks, it is crucial to consider that most IoT devices face severe limitations in terms of their data acquisition capabilities, computational power, and memory size. Hence, the successful implementation of efficient NN-based algorithms that can handle big data while considering the resource limitations of IoT devices is a critical challenge to address. In this paper, we propose a novel denoiser for a noisy EDM, referred to as DAE-EDMR (Readers can understand our framework more easily by referring to Figure 1. Ourmethod is based on the mathematical fact that the EDM measured from \(N\) nodes in a \(k\)-dimensional space must have a rank of at most \(k+2\). Abstractly, this implies that \(N\)-choose-2 pairwise distances can be rearranged into a \(k\)-dimensional structure. Consequently, if the pairwise distances are not accurately measured, reconstructing these segments would require embedding them into a higher-dimensional space. Using this concept, we optimize the NN model for EDM denoising by utilizing the eigenvalues to capture how the nodes are volumetrically distributed. A detailed explanation is given in Section 2), which leverages a denoising autoencoder (DAE). To increase the efficiency of the NN operations, we exploit the low-rank property of the EDM, which is bounded to only a \(k+2\) number of eigenvalues, where \(k\) is the dimension of the Euclidean space, independent of the number of sensor nodes [23]. Leveraging this valuable mathematical observation, we design our NN model by inputting and outputting vectors of the eigenvalues of the noisy and original EDMs, respectively. Through these inputting/outputting rules, the proposed scheme achieves remarkable denoising results, even with a relatively small training dataset. The efficiency of the proposed denoiser is attributed to the utilization of the low-rank property of the EDM, which helps the NN model to establish better inference with limited training data. In fact, the combination of the DAE and EDM is highly attractive due to the complementary features of these two techniques. On the one hand, the DAE is a powerful NN framework for manifold learning, which enables it to effectively capture intricate data patterns and facilitate efficient denoising. On the other hand, the EDM exhibits an extremely-low-rank property, contributing to the dimensionality reduction.