## 2 The Proposed Schemes: DAE-EDMR and T-DAE-EDMR
### Method II: Truncated DAE-EDMR (T-DAE-EDMR)  
This subsection introduces T-DAE-EDMR, which is the relaxed version of DAE-EDMR. From the previous subsection, the low-rank property of the EDM can be used more efficiently from the NN training/test point of view. Assume that the NN model is optimized in a network with \(N\) nodes. Here, we consider a scenario where the number of nodes changes at the test phase, which is frequently shown in wireless networks. If the NN model can be used flexibly under this kind of environment change, it will be a more efficient utilization. For this reason, we newly define the optimal denoising function \(\mathcal{H}_{T}^{*}\): \(\mathbb{R}^{k+2}\mapsto\mathbb{R}^{k+2}\), which can be formulated as follows:  
\[\mathcal{H}_{T}^{*}=\operatorname*{argmin}_{\mathcal{H}}||\mathbf{s}-\mathcal{ H}(\tilde{\mathbf{s}}_{T})||_{2}^{2}, \tag{15}\]  
where \(\tilde{\mathbf{s}}_{T}\in\mathbb{R}^{k+2}\) is the \((k+2)\)-truncated vector of \(\tilde{\mathbf{s}}\) in descending order. Again, we can design T-DAE-EDMR denoted by \(\mathcal{H}_{T}^{\text{NN}}\) with a new \(\{f_{\theta*},g_{\theta^{*}}\}\) as \(\{f_{\theta_{T}*},g_{\theta_{T}*}\}\). In order to construct T-DAE-EDMR, we only need to change the dimension of the weight matrix for encoding to \(\mathbf{W}\in\mathbb{R}^{N^{\prime}\times(k+2)}\), and all the other configurations are the same as for \(\mathcal{H}^{\text{NN}}\). Now, \(\mathcal{H}_{T}^{\text{NN}}\) can be obtained through the \(M\) training dataset as follows:  
\[\mathcal{H}_{T}^{\text{NN}}=\{f_{\theta_{T}*},g_{\theta_{T}*}^{\theta_{T}*}\}= \operatorname*{argmin}_{f_{\theta_{T}*}g_{\theta^{*}}}\frac{1}{M}\sum_{i=1}^{M }||\mathbf{s}^{(i)}-g_{\theta^{\prime}}(f_{\theta}(\tilde{\mathbf{s}}_{T}^{(i) }))||_{2}^{2}. \tag{16}\]  
Next, in the test phase, let \(\hat{\mathbf{s}}_{T}^{(j)}\) denote the \(j\)-th denoised eigenvalues with the optimized \(\mathcal{H}_{T}^{\text{NN}}\), and this can be written as  
\[\hat{\mathbf{s}}_{T}^{(j)}=\mathcal{H}_{T}^{\text{NN}}(\hat{\mathbf{s}}_{T}^{( j)})=g_{\theta_{T}^{*}}(f_{\theta_{T}*}(\hat{\mathbf{s}}_{T}^{(j)})). \tag{17}\]  
Finally, in the test phase, we can again obtain the \(j\)-th denoised EDM \(\hat{\mathbf{D}}_{T}^{(j)}\) as follows:  
\[\hat{\mathbf{D}}_{T}^{(j)}=\tilde{\mathbf{V}}^{(j)}\text{diag}(\hat{\mathbf{s }}_{T}^{(j)})\big{[}\hat{\mathbf{V}}^{(j)}\big{]}^{T}. \tag{18}\]After obtaining \(\hat{\mathbf{D}}_{T}\), performing connectivity-based localization involves repeating the work in Section 2.2.2 but replacing \(\hat{\mathbf{D}}\) with \(\hat{\mathbf{D}}_{T}\). Overall, Algorithms 1 and 2 describe the processes of DAE-EDMR and T-DAE-EDMR, respectively. ```
1:[The training phase (\(M\): number of training datasets)]
2:Collect the training dataset of true and noisy EDMs, i.e., \(\mathbf{D}^{(i)}\) and \(\hat{\mathbf{D}}^{(i)}\), for all \(i\in\{1,\cdots,M\}\). 3:for\(i\gets 1\) to \(M\), do
4: Extract the vectors of the eigenvalues of \(\mathbf{D}^{(i)}\) and \(\hat{\mathbf{D}}^{(i)}\), i.e., \(\mathbf{s}^{(i)}\) and \(\hat{\mathbf{s}}^{(i)}\). 5:endfor
6:Optimize the NN-based denoiser consisting of encoding/decoding functions based on (9), i.e., \(\mathcal{H}^{\text{NN}}(=\{f_{\theta*},g_{\theta*}\})\), by inputting and outputting \(\hat{\mathbf{s}}^{(i)}\) and \(\mathbf{s}^{(i)}\), respectively. 7:[The test phase (\(L\): number of test datasets)]
8:Collect the test dataset of true and noisy EDMs, i.e., \(\mathbf{D}^{(i)}\) and \(\tilde{\mathbf{D}}^{(i)}\), for all \(j\in\{1,\cdots,L\}\). 9:for\(j\gets 1\) to \(L\), do
10:Make the input vector \(\hat{\mathbf{s}}^{(j)}\) referring to step 4. 11:Generate \(\hat{\mathbf{s}}^{(j)}\) by passing \(\hat{\mathbf{s}}^{(j)}\) to \(\mathcal{H}^{\text{NN}}\). 12:Obtain the denoised EDM, i.e., \(\hat{\mathbf{D}}^{(i)}\), based on (12). 13:Implement the classical MDS with \(\hat{\mathbf{D}}^{(j)}\) to obtain the estimate of \(\mathbf{X}^{(j)}\).