esa-sceva/satcom-chunk-collection huggingface dataset:
## 5 Application to Manufacturing System Design Problems
### Production Rate Maximization Problem
The number in parenthesis below \(z_{i}\) denotes the number of the buffer positions of group \(i\), and the maximum value of the TB was 174 when all \(z_{i}=UB_{i}\). The optimal buffer allocations obtained from CPLEX(r) were not always global optimal. Figure 4 shows the behavior of the production rate and relative optimal buffer allocation as TB increases. In this figure, reconfiguration strategy 3 always performs better than strategies 1 and 2. However, the gaps are negligible when the TB is greater than 150. The production rates obtained from the meta-model of strategy 3 were compared with those from simulation, as shown in Figure 5. The buffer allocations were optimized using the meta-model, and the same values were applied to simulation model. The detail optimization results are listed in Table 9. The meta-model overestimated the production rates when the TB was small; however, it changed to being underestimated when the TB increased. The gap between the two methods reduced in the middle of the ranges of the  
Figure 4: Behavior of production rate with total buffer (TB) and reconfiguration strategies.
## 4 Experimental Study and Performance Benchmarking  
To measure the performance of the algorithms outlined above, these were used for the MMO of multiple PFSSP. Eighteen benchmark problems from various datasets were taken for experiments. The first nine problems were proposed by [PERSON] [35], the next eight were proposed by [PERSON] [36] and the last one was proposed by [PERSON] [37]. To better assess the algorithms, 10 independent simulations were performed for each test problems. The performance of the algorithms used was then measured on the basis of four indicators. (1) The best solution found in the 10 simulations. (2) The mean value  
Figure 7: Clearing method procedure [33].  
and the standard deviation of the optimal solution found for the 10 simulations. (3) The number of times the algorithms converged to the best optimal solution. (4) The average number of best optimal solutions found for the 10 simulations. For indicator number 4, only simulations in which the algorithm converged to the best solution from (1) were taken into consideration i.e., if the algorithm only converged to the best solution in 4 out of 10 simulations, then the average number of optimal solutions found was calculated using those 4 simulations. The parameters used for the three algorithms were determined after numerous simulations and are given in Table 1. Additionally, the performance of the three algorithms was also compared to the performance of Improved Efficient Genetic Algorithm (IEGA) and the Hybrid IEGA (HIEGA) proposed by [PERSON] et al. [38], when applicable. These algorithms were selected as they showed promising results when used for the optimization of benchmark PFSSPs.
## 2 Introduction to Related Methods
### Bidirectional Gated Recurrent Unit (Bi-GRU)  
The Bi-GRU network comprises two opposing Gated Recurrent Units (GRUs) [17], facilitating both forward and backward propagation to simultaneously harness historical and future information from the data. By collecting vibration signals from bearings for life prediction, the results are associated with historical feature information and the overall trend of changes. The Bi-GRU model effectively captures long-term dependencies, thereby enabling more precise predictions of the bearing's RUL. A schematic diagram of the Bi-GRU model structure is shown in Figure 3.  
Figure 2: Structural diagram of the multi-dimensional hybrid gated attention mechanism.  
The calculation formula for the update gate of the Gated Recurrent Unit (GRU) is as follows:  
1. The input \(X_{n}\) at time step \(n\) is concatenated with the hidden state \(\text{h}_{n-1}\) from the previous time step \(n-1\), followed by a linear transformation. The result is then passed through a sigmoid function to output values within the fixed range of 0 to 1. \[Z_{n}=\sigma(W_{n}*(h_{t-1},x(n)))\] (5) where \(X_{n}\) is the input, \(Y_{n}\) is the output, \(W_{n}\) represents learnable parameters, \(\sigma\) denotes the sigmoid activation function, \(Z_{n}\) stands for the update gate, and \(h_{t-1}\) represents the hidden information from the previous time step.
2. The reset gate primarily determines how much historical information should be forgotten. \[r_{n}=\sigma(W_{r}*(h_{t-1},x(n)))\] (6) where \(r_{n}\) is the reset gate and \(W_{r}\) is the parameter matrix of the reset gate.
3. The reset gate is multiplied by the hidden state output \(Y_{n-1}\) from the previous time step \(n-1\). This product determines how much of the information from the previous step is retained. The result is then passed through the \(\tanh\) activation function to produce the new output \(Y_{n}\). \[b_{n}=\tan(W_{t}*[r_{n}\times h_{t-1},x(n)])\] (7) where \(W_{t}\) is a trainable weight matrix.
4. The gate value output by the update gate \((1-Z_{n})\) is multiplied by the hidden state output from the previous time step \((n-1)\) to determine the information to be retained. Finally, the result is added to obtain the final hidden state output \(Y_{n}\). \[Y_{n}=(1-Z_{n})*Y_{n-1}+Z_{n}*b_{n}\] (8) The mathematical expressions for the BiGRU network architecture are as follows: \[\overset{\leftarrow}{Y}_{n}=G_{RUI}(x_{n},\overset{\leftarrow}{Y}_{n-1})\] (9) \[\overset{\rightarrow}{Y}_{n}=G_{RUI}(x_{n},\overset{\rightarrow}{Y}_{n-1})\] (10) where \(\overset{\rightarrow}{Y}_{n}\) denotes forward propagation and \(\overset{\leftarrow}{Y}_{n}\) denotes backward propagation.  
Figure 3: Bi-GRU architecture diagram.
# Machine Learning-Based Satellite Routing for SAGIN IoT Networks
## 4 Experimental Simulation Results and Analysis
### Metrics of Extended ELM Model  
To characterize the effectiveness of the ELM classification method, the confusion matrix is used to evaluate the performance of the method [23]. The confusion matrix combines accuracy, precision, recall, and F1-score to measure the performance of the classification model. It records the sample number of all classification results in the table. Accuracy, precision, recall, and F1-score are defined as follows:  
\[\mathrm{Accuracy}=\frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{TP}+\mathrm{TN}+ \mathrm{FP}+\mathrm{FN}} \tag{15}\]  
\[\mathrm{Precision}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}} \tag{16}\]  
\[\mathrm{Recall}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}} \tag{17}\]  
\[\mathrm{F1}-\mathrm{score}=\frac{2\mathrm{TP}}{2\mathrm{TP}+\mathrm{FP}+ \mathrm{FN}} \tag{18}\]  
In the formulas, TP, TN, FP, and FN represent true positive, true negative, false positive, and false negative respectively. In the process of evaluating the model, if only one index of accuracy or recall rate is used, the advantages and disadvantages of the model cannot be fully evaluated. Therefore, the F1 score, as a harmonic average of accuracy and recall rate, can be better used as an evaluation index of model of classification in multi-classification problems.
# Dual Histogram Equalization Algorithm Based on Adaptive Image Correction
## 4 Analysis of Algorithm Results
### Objective Evaluation Indicators
#### 4.3.3 Absolute Mean Brightness Error  
The absolute average brightness error (_AMBE_) indicates the absolute difference between the average brightness of the input image and the resulting image, and it is used to measure the performance in maintaining the original brightness. ASME is a value greater than zero. The smaller the _AMBE_ value, the better the light preservation effect. It is calculated as follows:  
\[AMBE=\mid Q(X)-Q(Y)\mid \tag{25}\]  
where \(X\) and \(Y\) denote the input image and the resultant image, respectively, and \(Q(X)\) and \(Q(Y)\) are the average brightness values of the input image and the resultant image, respectively.
## 4 Distributed BS Assignment Algorithm  
The proposed BS assignment algorithm is based on the supply and demand scheme, where prices per resource unit are defined based on the information exchange between mobile devices and BSs. Subsequently, a bidding process is proposed in order to steer the BS assignment depending on the prices defined by each BS. The bidding process in our proposed solution is realized using two algorithms, one running on the mobile device and another one at the BS. Under this pricing framework, and the mapping of the original optimization problem into the dual problem, each BS \(j\) computes the prices defined in Equations (10) and (11). The underlying idea of our approach is that each device selects the serving BS based on the weighted utility. This, in turn, depends on the prices that are disseminated by BSs, which are updated as function of the demand experienced at each BS.
## 6 Proposed Approach  
Our proposed approach consists of two phases, i.e., area coverage optimization and energy consumption optimization. To achieve high coverage ratio, low redundancy ratio and energy consumption balance, we utilize an improved adaptive particle swarm optimization (IAPSO) to solve multi-objective area coverage optimization model and cluster head selection optimization model.
## 3 Results
### Characterization of Mesoporous Silica
#### 3.1.2 Transmission Electron Microscopy Measurements  
The type and pore distribution of the materials were evaluated by TEM, as illustrated in Figure 4. TEM micrographs of the MCM-41 sample exhibit a well-defined and ordered porous structure, showing a parallel arrangement of cylindrical pore channels in the [100] direction (Figure 4a) and a hexagonal honeycomb-like arrangement along the [001] axis (Figure 4a inset) [34]. On the other hand, the less ordered structure of the MSU-2 silica, with a wormhole-like pattern, is illustrated by the TEM micrographs in Figure 4b. These micrographs show that the silica particle contains a large number of channels that are regular in diameter, although they lack long-range packing order. Furthermore, TEM confirms that the entire material is composed of this porous structure, with no non-porous components present, and reveals the spherical shape of the particles [35].  
Figure 3: X-ray diffraction (XRD) patterns of MCM-41 (green color) and MSU-2 (blue color).
## 3 Results
### Influence of Dynamism of the DVFs on Emotion Recognition (H2)  
In regards to the effect of dynamism on the ability to recognize emotions among both patients and healthy controls, different results were found. The Wilcoxon signed-rank test showed that the disparities in the emotion recognition rates were statistically significant; these rates were greater for the most dynamic DVFs (\(Z=-3.392\), \(p=0.001\)). Nevertheless, it was found that the patient group did not exhibit a greater degree of accuracy in recognizing the most dynamic virtual faces, as compared to the less dynamic ones (\(Z=-1.114\), \(p=0.265\)).  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**MDD Group** & **Neutral** & **Surprise** & **Fear** & **Anger** & **Disgust** & **Joy** & **Sadness** \\ \hline Neutral & 90.3\% & 2.8\% & 0.9\% & 1.4\% & 0.5\% & 0.5\% & 3.7\% \\ Surprise & 2.3\% & 89.6\% & 4.4\% & 1.4\% & 0.2\% & 0.9\% & 1.2\% \\ Fear & 1.2\% & 41.7\% & 48.4\% & 2.8\% & 1.9\% & 0.2\% & 3.9\% \\ Anger & 2.1\% & 5.1\% & 2.5\% & 83.6\% & 5.3\% & 0.0\% & 1.4\% \\ Disgust & 1.2\% & 7.2\% & 4.2\% & 19.9\% & 66.9\% & 0.2\% & 0.5\% \\ Joy & 6.0\% & 4.2\% & 0.9\% & 1.4\% & 2.3\% & 84.5\% & 0.7\% \\ Sadness & 7.9\% & 7.6\% & 8.1\% & 8.1\% & 4.4\% & 0.9\% & 63.0\% \\ \hline
**Healthy group** & Neutral & Surprise & Fear & Anger & Disgust & Joy & Sadness \\ \hline Neutral & 94.0\% & 0.5\% & 0.5\% & 0.0\% & 0.9\% & 0.0\% & 4.2\% \\ Surprise & 0.9\% & 90.3\% & 8.3\% & 0.0\% & 0.0\% & 0.0\% & 0.5\% \\ Fear & 0.9\% & 12.7\% & 77.3\% & 0.2\% & 0.7\% & 0.0\% & 8.1\% \\ Anger & 0.7\% & 1.2\% & 1.9\% & 92.4\% & 3.0\% & 0.0\% & 0.9\% \\ Disgust & 0.2\% & 0.5\% & 0.9\% & 13.2\% & 85.0\% & 0.0\% & 0.2\% \\ Joy & 4.9\% & 0.7\% & 0.2\% & 0.7\% & 0.5\% & 93.1\% & 0.0\% \\ Sadness & 3.0\% & 3.5\% & 5.6\% & 0.9\% & 1.6\% & 0.0\% & 85.4\% \\ \hline \hline \end{tabular} Columns: Recognized emotions. Rows: Displayed emotions.  
\end{table}
Table 2: Emotion recognition scores for each emotion depicted for the depression group and healthy controls.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Neutral** & **Surprise** & **Fear** & **Anger** & **Disgust** & **Joy** & **Sadness** \\ \hline
**MDD group** & 6.25 (3.38) & 3.93 (1.61) & 4.55 (1.63) & 4.59 (2.87) & 4.55 (1.56) & 4.47 (2.97) & 5.41 (2.75) \\
**Healthy group** & 2.85 (1.24) & 2.73 (1.16) & 2.53 (1.02) & 2.58 (1.14) & 2.46 (1.04) & 2.25 (0.84) & 2.16 (0.77) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Average reaction time and standard deviation (in seconds) per emotion for the depression and healthy groups.
## 2 Methods  
### Wave Mode Decomposition Theory  
For two-dimensional (2D) isotropic elastic media, the coupled elastic seismic wavefields \((u_{x},u_{z})\) can be decomposed into the horizontal and vertical components of P- and S-wave modes \((u_{xP},u_{zP},u_{xS},u_{zS})\), which can be expressed as  
\[u_{x}=u_{xP}+u_{xS},u_{z}=u_{zP}+u_{zS} \tag{1}\]The essence of vector decomposition of P- and S-wave modes is to redistribute the energy from the coupled elastic seismic wavefields based on the waveform differences of different wave modes, which can be theoretically regarded as a point-to-point prediction problem. To effectively solve this nonlinear prediction problem, deep convolutional generative adversarial networks (DCGANs) [40] can be used to develop the intelligent data-driven vector decomposition method of P- and S-wave modes. In other words, the vector decomposition method of different wave modes based on DCGANs belongs to an intelligent image processing algorithm and therefore is not dependent on elastic model parameters and certain prior conditions compared with the conventional algorithms. According to Equation (1), the goal of vector decomposition is to accurately separate four decoupled wavefield components from two coupled wavefield components using DCGANs, and therefore its network structure should ideally have two channels for inputting the horizontal and vertical components of coupled elastic seismic wavefields (\(u_{x},u_{z}\)) and four channels for outputting the horizontal and vertical components of the decoupled vector P- and S-wave modes (\(u_{xP},u_{zP},u_{xS},u_{zS}\)). However, the network structure is relatively complex according to this above strategy which may lead to an unstable training process, and therefore we use the workflow shown in Figure 1 for the vector decomposition of P- and S-wave modes to obtain better training results. From Figure 1, the network structure of DCGANs has only two channels to output the horizontal and vertical components of the decoupled vector P-wave mode (\(u_{xP},u_{zP}\)) after predicting the wave mode of each point for input samples, which can be expressed as  
\[(u_{xP},u_{zP})=f[(u_{x},u_{z});\theta] \tag{2}\]  
where \(f(\cdot)\) is the DCGANs, and \(\theta\) represents its network hyperparameters. By reducing the output channels of the neural network, we can simplify the network structure as much as possible and further stabilize the training process. The horizontal and vertical components of the decoupled vector S-wave mode can finally be obtained through the strategy of wavefield subtraction, which can be written as  
\[u_{xS}=u_{x}-u_{xP},u_{zS}=u_{z}-u_{zP} \tag{3}\]  
Figure 1: The workflow for the vector decomposition of P- and S-wave modes from the coupled elastic seismic wavefields.
# Exploring the Impact of Multitemporal DEM Data on the Susceptibility Mapping of Landslides
## 4 Results
### Susceptibility Assessments of Landslides  
The susceptibility assessment maps of the study area in 1992, 2000, and 2009 are obtained in the GIS environment using LRM (Figure 8 and Table 4). The susceptibility is classified into four levels based on the natural breaks classification (NBC) method, which is based on natural groupings inherent in data. The NBC identifies groups of similar values and maximizes the differences between classes. The features are divided into classes whose boundaries are set based on relatively big differences in the data values. Here, level I, level II, level III, and level IV denote low, moderate, high, and very high susceptibility, respectively. The area percentages of various susceptibility levels in the study are shown in Table 4.  
As shown in Figure 8 and Table 4, the results of the susceptibility assessment vary from year to year. The areas with low and moderate susceptibility (levels I and II, respectively) in 1992 are the largest, while the area with very high susceptibility (level IV) is the largest in 2000. The rapid decrease in the level IV area from 2000 to 2009 is noticeable. One possible explanation for the decrease is that the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Susceptibility**} & \multicolumn{2}{c}{**1992**} & \multicolumn{2}{c}{**2000**} & \multicolumn{2}{c}{**2009**} \\ \cline{2-7}  & **Area** & **Percentage** & **Area** & **Percentage** & **Area** & **Percentage** \\  & **(10\({}^{3}\) km\({}^{2}\))** & **(\%)** & **(10\({}^{3}\) km\({}^{2}\))** & **(\%)** & **(10\({}^{3}\) km\({}^{2}\))** & **(\%)** \\ \hline Level I & 26.445 & 34.23 & 22.342 & 28.91 & 23.378 & 30.21 \\ Level II & 26.127 & 33.82 & 17.969 & 23.25 & 15.781 & 20.39 \\ Level III & 13.573 & 17.57 & 17.128 & 22.16 & 20.747 & 26.81 \\ Level IV & 11.116 & 14.39 & 19.842 & 25.68 & 17.473 & 22.58 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Susceptibility assessment of landslides using LRM scores.  
Figure 8: Susceptibility assessment maps for (**a**) 1992, (**b**) 2000, and (**c**) 2009.  
slopes where landslides occurred became stable in the short term, owing to the excessive landslides that occurred from 2000 to 2009; thus, the susceptibility level of the study area decreased after the hazards.  
The susceptibility maps are obtained using different DEM-dependent factors, while the DEM-independent factors are constant in the assessment. Comparing these results illustrates the significant influence of the DEM data from different years on the susceptibility assessment.
## 5 Theoretical Analyses
### Optimality of the Discriminator  
**Proposition 1**.: _For a given generator \(G\), the optimal discriminator \(D\) is as follows:_  
\[\mu_{D}^{*}(x)=\frac{1}{1+\exp(-\frac{p_{d}(x)-p_{g}(x)}{\beta(p_{d}(x)+p_{g}( x))})}=\sigma(\frac{p_{d}(x)-p_{g}(x)}{\beta(p_{d}(x)+p_{g}(x))}) \tag{8}\]  
_where \(\sigma(\cdot)\) is the sigmoid function._  
Proof.: According to Equations (4) and (5), for a given generator \(G\), the training criterion for the discriminator \(D\) is to maximize the following value function marked as \(J\):  
\[J=\int_{x}p_{d}(x)[\mu_{D}(x)+\beta\mathrm{H}_{D}(x)]\mathrm{d}x+\int_{x}p_{g}( x)[\mu_{\
eg D}(x)\mathrm{d}x+\beta\mathrm{H}_{\
eg D}(x)]\mathrm{d}x\]  
Let \(F\) be the following form:  
\[F=p_{d}(x)[\mu_{D}(x)+\beta\mathrm{H}_{D}(x)]+p_{g}(x)[\mu_{\
eg D}+\beta \mathrm{H}_{\
eg D}(x)]\]  
Then, \(J=\int_{x}F\mathrm{d}x\). Combined with Equation (3) and \(\mu_{\
eg D}(x)=1-\mu_{D}(x)\), the first and second derivatives of \(F\) with respect to \(\mu_{D}(x)\) are as follows:  
\[\frac{\partial F}{\partial\mu_{D}(x)}=p_{d}(x)-p_{g}(x)-\beta[p_{d}(x)+p_{g}( x)][\ln\mu_{D}(x)-\ln\mu_{\
eg D}(x)] \tag{9}\]  
\[\frac{\partial^{2}F}{\partial\mu_{D}(x)^{2}}=\frac{\beta(p_{d}(x)+p_{g}(x))}{ \mu_{D}(x)^{2}-\mu_{D}(x)} \tag{10}\]  
Obviously, Equation (10) is less than 0 since \(\mu_{D}(x)\in[0,1]\). Therefore, the optimal solution of \(J\) can be obtained by setting Equation (9) = 0. Consequently, we can obtain the optimal discriminator \(D\), as shown in Equation (8).  
Next, we analyze the discriminator \(D\) qualitatively. According to Equation (8), \(\mu_{D}(x)=0.5\) when \(p_{d}(x)\to 0\) and \(p_{g}(x)\to 0\). For the interval where \(p_{d}(x)\geq p_{g}(x)\), the value of \(\mu_{D}(x)\in[0.5,1]\), which is negatively correlated with the Earth Mover's Distance between \(p_{d}(x)\) and \(p_{g}(x)\). The larger the distance is, the smaller the \(\mu_{D}(x)\) is, and the smaller the distance is, the larger the \(\mu_{D}(x)\) is. As the training progresses, \(P_{g}\) moves in the direction of \(P_{d}\), and the Earth Mover's Distance between the two distributions decreases gradually. Correspondingly,the curve of the \(\mu_{D}(x)\) also becomes flatter and flatter. When the training is over, \(\mu_{D}(x)=0.5\) in the entire domain. The schematic diagram of this process is shown in Figure 2.
# Fast Normalization for Bilinear Pooling via Eigenvalue Regularization
## 3 Our Approach
### First Stage  
Despite being powerful, the representation \(\mathbf{A}\) lies in the Riemannian manifold. Obviously, it is not adapted to a linear classifier (fully connected layer), which works in the Euclidean space. Hence, at the first stage, we perform square root normalization on the \(\mathbf{A}\) before the classifier. The solution normalizes \(\mathbf{A}\) into its square-rooted one: \(\mathbf{A}^{1/2}\) so that \(\mathbf{A}=\mathbf{A}^{1/2}\mathbf{A}^{1/2}\).  
Originally, the normalization can be processed with the SVD [22; 57]. Let the SVD of \(\mathbf{A}\) be expressed as  
\[\mathbf{A}=\mathbf{Q}\mathbf{\Sigma}\mathbf{Q}^{T} \tag{2}\]  
where \(\mathbf{Q}^{D\times D}\) is the matrix of eigenvectors, and the diagonal matrix \(\mathbf{\Sigma}\in\mathbb{R}^{D\times D}\) contains eigenvalues. The normalized matrix is obtained by square rooting diagonal elements in \(\mathbf{\Sigma}\):  
\[\mathbf{A}^{1/2}=\mathbf{Q}\mathbf{\Sigma}^{1/2}\mathbf{Q}^{T}. \tag{3}\]  
Nevertheless, computing square-rooted matrix \(\mathbf{A}^{1/2}\) via SVD is less efficient on the GPU and slows the whole training process. Alternatively, at the first stage, we use [PERSON]'s iteration, as proposed in [15; 20], to calculate approximated \(\mathbf{A}^{1/2}\). Specifically, it is dedicated to solving the formula \(F(\mathbf{Z})=\mathbf{Z}^{2}-\mathbf{A}=0\). The approach is an iterative process where each iteration is as follows:  
\[\mathbf{Y}_{k+1}=\frac{1}{2}\mathbf{Y}_{k}(3\mathbf{I}-\mathbf{Z}_{k}\mathbf{ Y}_{k}), \tag{4}\]  
\[\mathbf{Z}_{k+1}=\frac{1}{2}(3\mathbf{I}-\mathbf{Z}_{k}\mathbf{Y}_{k})\mathbf{ Z}_{k}. \tag{5}\]  
As illustrated in Figure 3, the inputs are \(\mathbf{Y}_{0}=\mathbf{A}\) and \(\mathbf{Z}_{0}=\mathbf{I}\). After \(K\) iterations, \(\mathbf{Y}_{K}\) and \(\mathbf{Z}_{K}\) converge to \(\mathbf{A}^{1/2}\) and \(\mathbf{A}^{-1/2}\). Like previous works, \(K\) is set to be 5 throughout the paper. Furthermore, Equations (4) and (5) indicate that the normalization process only consists of matrix multiplications which are well supported on the GPU. Therefore, we adopt it at the first stage to speed up the normalization.  
Figure 3: Square root normalization with [PERSON]’s iteration. The inputs are a bilinear representation matrix \(\mathbf{A}\) and an identity matrix \(\mathbf{I}\). After \(K\) iterations, the outputs \(\mathbf{Y}_{n}\) and \(\mathbf{Z}_{n}\) are, respectively, the approximation of the square-rooted matrix \(\mathbf{A}^{1/2}\) and of the inverse square-rooted matrix \(\mathbf{A}^{-1/2}\). Each iteration corresponds to Equations (4) and (5).  
Finally, the normalized representation \(\mathbf{A}^{1/2}\) is flattened as a vector and sent to a linear classifier. The network is trained with the cross-entropy loss \(L_{CE}\):  
\[\mathcal{L}_{CE}=-\frac{1}{S}\sum_{s}^{S}\sum_{c}^{C}q_{c}^{\text{s}}\text{log} (p_{c}^{\text{s}}), \tag{6}\]  
where \(q_{c}^{\text{s}}\) and \(p_{c}^{\text{s}}\) are, respectively, the target and predicted probability values at \(c\)-class, respectively, of the \(s\)-th training sample.