## 2 The Proposed Schemes: DAE-EDMR and T-DAE-EDMR
### Method I: DAE-EDMR
#### 2.2.1 A Denoising Process  
Before describing the framework of DAE-EDMR, we will revisit the low-rank property of an EDM [23]. **Property 1**. : _The rank \((\mathbf{D})\), corresponding to the points in \(\mathbb{R}^{k}\), is at most \(k+2\)._  
This is based on the following equality:  
\[\mathbf{D}=\mathbf{1}_{N}\text{diag}(\mathbf{X}^{T}\mathbf{X})^{T}-2\mathbf{ X}^{T}\mathbf{X}+\text{diag}(\mathbf{X}^{T}\mathbf{X})\mathbf{1}_{N}^{T}, \tag{7}\]where \(\mathbf{1}_{N}\) is the one-vector of size \(N\). According to rank characteristics, the rank of \(\mathbf{D}\) is bounded to the summation of the ranks of each term, i.e., \(\text{rank}(\mathbf{D})\leq\text{rank}(\mathbf{1}_{N}\text{diag}(\mathbf{X}^{ T}\mathbf{X})^{T})+\text{rank}(\mathbf{X}^{T}\mathbf{X})+\text{rank}(\text{diag}( \mathbf{X}^{T}\mathbf{X})\mathbf{1}_{N}^{T})=1+k+1\). This determines the rank of an EDM as extremely low, i.e., \(\text{rank}(\mathbf{D})\leq k+2\), regardless of the number of nodes. It implies that it is effective to perform denoising given the small dimension of the (potential) latent space of an NN model, i.e., the similar level of the rank of the EDM, rather than treating the distance information as a whole. Now, let \(\mathbf{s}\in\mathbb{R}^{k+2}\) and \(\mathbf{\hat{s}}\in\mathbb{R}^{N}\) denote the vectors whose elements are the descending-order eigenvalues of \(\mathbf{D}\) and \(\mathbf{\hat{D}}\), respectively. Thus, the optimal denoising function \(\mathcal{H}^{*}\): \(\mathbb{R}^{N}\mapsto\mathbb{R}^{k+2}\) can be constructed as follows:  
\[\mathcal{H}^{*}=\underset{\mathcal{H}}{\text{argmin}}\,||\mathbf{s}-\mathcal{ H}(\mathbf{\hat{s}})||_{2}^{2}. \tag{8}\]  
Here, we design a fully connected NN framework denoted by \(\mathcal{H}^{\text{NN}}\) to approximate \(\mathcal{H}^{*}\). To this end, we will define the required terms as follows:  
* \(N^{\prime}\): The dimension of the latent space. * \(\mathbf{W}\in\mathbb{R}^{N^{\prime}\times N},\mathbf{W}^{\prime}\in\mathbb{R} ^{(k+2)\times N^{\prime}}\): The weight matrices for encoding and decoding, respectively. * \(\mathbf{b}\in\mathbb{R}^{N^{\prime}},\mathbf{b}^{\prime}\in\mathbb{R}^{k+2}\): The bias vectors for encoding and decoding, respectively. * \(\mathcal{S}\): The activation function for neural networks. At the propagation between the final hidden layer and the output layer, \(\mathcal{S}(a)=a\), i.e., an identity function. For other types of propagation between adjacent layers, \(\mathcal{S}(a)=\frac{\ell^{a}-\ell^{a}}{\ell^{a}+\ell^{a}}\), i.e., a hyperbolic tangent function. And \(\mathcal{S}(\mathbf{a})=(\mathcal{S}(\mathbf{a}[1]),\cdots,\mathcal{S}( \mathbf{a}[P]))^{T}\) where \(\mathbf{a}\in\mathbb{R}^{P}\) is an arbitrary input vector. With these terms (For simplicity, the description of the NN model design throughout this article is based on a single hidden layer; however, it is obvious that deeper hidden layers can be made using multiple encoding/decoding function parameters, i.e., \(\{\theta_{i}\}_{i=1}^{I}\) and \(\{\theta^{\prime}_{i}\}_{i=1}^{I}\), where \(I\), \(\theta_{i}\), and \(\theta^{\prime}_{i}\) are the depth of the NN model and the \(i\)-th encoding and decoding function parameters), we define \(f_{\theta}\) as the encoding function where the parameter \(\theta\) is \(\{\mathbf{W},\mathbf{b}\}\), i.e., \(f_{\theta}(\mathbf{\hat{s}})=\mathcal{S}(\mathbf{W}\mathbf{\hat{s}}+\mathbf{b})\). In addition, we define \(g_{\theta^{\prime}}\) as the decoding function where the parameter \(\theta^{\prime}\) is \(\{\mathbf{W}^{\prime},\mathbf{b}^{\prime}\}\), i.e., \(g_{\theta^{\prime}}(f_{\theta}(\mathbf{\hat{s}}))=\mathcal{S}(\mathbf{W}^{ \prime}f_{\theta}(\mathbf{\hat{s}})+\mathbf{b}^{\prime})\).