esa-sceva/satcom-chunk-collection huggingface dataset:
# Backscatter Assisted NOMA-PLNC Based Wireless Networks
## 5 Bit Error Rate Analysis
### Downlink--NOBC  
At destination, near node pair apply SIC operation, so it first decodes far node pair signal by applying zero forcing equalization.  
\[\hat{x}_{\tau_{2}}\;=\;\frac{z_{n_{i}}}{g_{i}};\quad i\in 1,2), \tag{44}\]  
\[\hat{x}_{\tau_{2}}\;\approx\;\alpha_{2}\sqrt{P_{r}}\hat{x}_{\tau_{2}}+in_{i}; \quad i\in 1,2), \tag{45}\]  
where \(in_{i}=\left(\alpha_{1}\sqrt{P_{r}}\hat{x}_{\tau_{1}}\;+\;\frac{n_{i}}{g_{i}}; \quad i\in 1,2\right)\),  
multiplying the detected symbol \(\hat{x}_{\tau_{2}}\) with its own message \(x_{3}\). Similarly, far node 2 estimate \(\hat{x}_{3}\). After removing far node pair signal, near node pair decode its own signal.  
\[\tilde{z}_{n_{i}}\;=\;(\hat{x}_{\tau_{3}}-\hat{x}_{\tau_{2}})g_{i}+n_{n_{i}} \quad i\in(1,2), \tag{46}\]  
\[\hat{x}_{\tau_{1}}\;\approx\;\alpha_{1}\sqrt{P_{r}}\hat{x}_{\tau_{1}}+\frac{n_ {i}}{g_{i}};\quad i\in 1,2), \tag{47}\]  
The instantaneous end-to-end BER of the proposed wireless network can be calculated for each node.  
\[p_{E-to-E_{i}}\;=\;p_{MAC}(1-p_{NOBC_{i}})+p_{NOBC_{i}}(1-p_{MAC});\quad i\in (1,2,3,4), \tag{48}\]  
where \(p_{NOBC_{i}}=Q(\sqrt{2\gamma_{i}^{d}}),\;i\in(1,2,3,4)\) is the instantaneous BER of each node and the average BER of each node is given as  
\[\chi_{i}(\overline{\gamma}_{i}^{d})=\left(1-\sqrt{\overline{\gamma}_{i}^{d}/( \overline{\gamma}_{i}^{d}+1)}\right)/2,\quad i\in(1,2,3,4), \tag{49}\]  
Then, the overall BER of the proposed wireless network can be computed as given by  
\[p_{E-to-E}=\sum_{n=1}^{4}p_{E-to-E_{i}}, \tag{50}\]  
Using (40), (48) and (49), the average overall end-to-end BER of the proposed wireless network can be computed as  
\[E[p_{E-to-E}]=2p_{A}+p_{B}-2(p_{C}+p_{D}+p_{E}+p_{F}), \tag{51}\]  
where  
\(p_{A}=\psi_{1}(\overline{\gamma}_{1}^{u})+\psi_{2}(\overline{\gamma}_{2}^{u })+\psi_{3}(\overline{\gamma}_{3}^{u})+\psi_{4}(\overline{\gamma}_{4}^{u})\), \(p_{B}=\chi_{1}(\overline{\gamma}_{1}^{d})+\chi_{2}(\overline{\gamma}_{2}^{d}) +\chi_{3}(\overline{\gamma}_{3}^{d})+\chi_{4}(\overline{\gamma}_{4}^{d})\),  
\(p_{C}=(\psi_{1}(\overline{\gamma}_{1}^{u})\chi_{1}(\overline{\gamma}_{1}^{d}) )+(\psi_{1}(\overline{\gamma}_{1}^{u})\chi_{2}(\overline{\gamma}_{2}^{d}))\), \(p_{D}=(\psi_{2}(\overline{\gamma}_{2}^{u})\chi_{1}(\overline{\gamma}_{1}^{d}) )+(\psi_{2}(\overline{\gamma}_{2}^{u})\chi_{2}(\overline{\gamma}_{2}^{d}))\),  
\(p_{E}=(\psi_{3}(\overline{\gamma}_{3}^{u})\chi_{3}(\overline{\gamma}_{3}^{u}) )+(\psi_{3}(\overline{\gamma}_{3}^{u})\chi_{4}(\overline{\gamma}_{4}^{u}))\), \(p_{F}=(\psi_{4}(\overline{\gamma}_{4}^{u})\chi_{3}(\overline{\gamma}_{3}^{u}) )+(\psi_{4}(\overline{\gamma}_{4}^{u})\chi_{4}(\overline{\gamma}_{4}^{u}))\).
## 5 Simulation Results
The proposed scheme applies both constructive and destructive SPC schemes, to improve the diversity gain of the mobile, and to mitigate interference signals. Case 2 in the conventional scheme does not satisfy the BER \(10^{-2}\) in all SNR environments. On the other hand, the proposed scheme increases the SNR of the desired signal and mitigates the interference through pre-coding scheme. So, the proposed scheme satisfies high BER performance in a low SNR environment. As a result, the proposed scheme can improve the performance and reliability of the HetNet systems by effectively mitigating interference signals even if the mobile is located between three adjacent pico cells. As shown in the simulation results, the proposed scheme has higher BER performance than the conventional scheme. The proposed scheme can effectively mitigate the interference by using the pre-coding in the HetNet system without the relay base station or relay mobile. As a result, the proposed scheme can improve the performance and reliability of the HetNet system.
## 3 Materials and Methods
### Field and Impedance Matching  
The field matching of the rectangular waveguide dominant TE\({}_{10}\) mode with the quasi TEM mode of microstrip line was achieved by varying the slot height, width, and back-short distance of the microstrip from the waveguide. The back-short distance was kept close to the quarter-guided wavelength at the design frequency to reduce reflections from the waveguide back-short end. This field-matching process is known as mechanical tuning. The impedance of the microstrip was 50 \(\Omega\), and that of the WR-28 was about 500 \(\Omega\). This significant difference in impedance was matched by varying the shape, size, and widthof the conductor strip probe inserted inside the waveguide through the waveguide slot, as well as through variation of the substrate width. Ultimately, a tapered fish-tail-shaped probe gave a better impedance matching for maximum EM energy transformation by varying the depth and width. This impedance-matching process is called electronic tuning.
# Performance Analysis of Two-Way Satellite-Terrestrial Relay Network with SWIPT
## 3 System Performance Analysis
### Average End-to-End Throughput of SPNC  
After obtaining the exact expressions for instantaneous error probability for PNC and SND protocols in the two-way satellite-terrestrial network for BPSK modulation, we can derive the instantaneous throughput of the SPNC protocol according to the throughput definition used in [27], the instantaneous end-to-end throughput of SPNC is given by  
\[Th_{\textit{spnc}}=\max(Th_{\textit{pnc}},Th_{\textit{snd}}) \tag{26}\]\[Th_{pmc}\simeq\frac{1}{2}(1-p_{MA}^{pmc})^{N}(1-p_{BC}^{pmc})^{N} \tag{27}\] \[Th_{snd}\simeq\frac{1}{2}(1-p_{MA}^{snd})^{N}(1-p_{BC}^{snd})^{N} \tag{28}\]  
where \(Th_{pmc}\) is the instantaneous of PNC protocol, \(Th_{snd}\) is the instantaneous of SND protocol. \(N\) is the number of bits long in each packet. Substituting Equations (12), (15), (17), and (18) into Equation (26), we can derive the instantaneous throughput of the SPNC protocol in case 1. Substituting Equations (17), (21), (23), and (25) into Equation (26), we can derive the instantaneous throughput of the SPNC protocol in case 2.  
Then the average end-to-end throughput of SPNC is written as:  
\[E\big{[}Th_{spnc}\big{]}=\int_{z^{pmc}}Th_{pmc}f(h,g)\mathrm{d}h\mathrm{d}g+ \int_{z^{snd}}Th_{snd}f(h,g)\mathrm{d}h\mathrm{d}g \tag{29}\]  
where \(f(h,g)\) is joint PDF of \(S_{1}\)--\(R\) and \(S_{2}\)--\(R\) link gain. \(z^{snd}\) is the \(z\)-plane region for BPSK defined in [8] written as \(z^{snd}=\left\{z\Big{|}\min(\sqrt{P_{2}}|g|,\sqrt{P_{1}}|h|)<\sqrt{W\sigma^{2} \ln(N/6)},|\mathcal{R}[z]|<0.5\right\}\). \(z^{pmc}\) is the complementary portion of \(z^{snd}\) in \(z\)-plane. \(\mathcal{R}[\cdot]\) denotes for real part operation.  
For the \(S_{1}\)--\(R\) and \(S_{2}\)--\(R\) link are independent; we can rewrite Equation (29) as:  
\[E\big{[}Th_{spnc}\big{]}=\int Th_{pmc}f(g)f(h)\mathrm{d}h\mathrm{d}g+\int_{z^{ snd}}(Th_{snd}-Th_{pmc})f(g)f(h)\mathrm{d}h\mathrm{d}g \tag{30}\]  
where \(f(h)\) is the PDF of \(S_{1}\)--\(R\) link, which follows Shadowed-Rician fading, \(f(g)\) is the PDF of \(S_{2}\)--\(R\) link, which follows Rayleigh fading.  
Substituting Equation (8) into (30), we can obtain the exact expressions of average end-to-end throughput for SPNC protocol. Furthermore, with the help of standard mathematical packets such as Mathematica, which is for numerical and symbolic operations, we can achieve the numerical computation of the confluent hypergeometric function \({}_{1}F_{1}(a;b;z)\). Then we can obtain the numerical results of average end-to-end throughput for SPNC protocol.
## 4 Simulation  
### HDS Channel Model  
We used the channel model in [16] to set up the HDS channel. The received signal \(b[j,k]\) was obtained from the HDS channel as follows:  
\[b[j,k]=a[j,k]\otimes h[j,k]+w[j,k], \tag{27}\]  
where \(h[j,k]\) is the point spread function (PSF) of the 2D ISI and \(\otimes\) is the 2D convolution operator. \(h[j,k]\) of the HDS channel can be calculated as follows:  
\[h[j,k]=\int\limits_{j-\frac{1}{2}}^{j+\frac{1}{2}}\int\limits_{k-\frac{1}{2}}^ {k+\frac{1}{2}}h(x,y)dxdy, \tag{28}\]  
where \(h(x,y)\) is the continuous PSF and defined as  
\[h(x,y)=\frac{1}{\sigma_{b}^{2}}\sin c^{2}\left(\frac{x-m_{x}}{\sigma_{b}}, \frac{y-m_{y}}{\sigma_{b}}\right); \tag{29}\]  
\(\sigma_{b}\) is the blur grade; \(m_{x}\) is the misalignments in the horizontal direction; \(m_{y}\) is the misalignments in the vertical direction; and \(\text{sinc}(x,y)=(\sin(\tau\text{x})/\tau\text{x})(\sin(\tau\text{y})/\tau \text{y})\). In this study, \(h[j,k]\) has a size of \(3\times 3\) array pixels. Finally, \(w[j,k]\) is the additive white Gaussian noise (AWGN) with zero mean and variance \(\sigma_{w}^{2}\). In addition, the SNR is defined as follows:  
\[\text{SNR}=10\log_{10}\left(\frac{1}{\sigma_{w}^{2}}\right), \tag{30}\]
## 2 Materials and Methods
### Method
#### 2.3.2 CARS-PLSR Feature Band Selection  
Spectral data typically contain a large number of highly correlated bands. This increases the complexity of the models [23; 24]. To address this issue, the Competitive Adaptive Reweighted Sampling (CARS) algorithm was introduced. The core idea of CARS is to simulate a natural selection process by adaptively adjusting the weights of each band and ultimately selecting the optimal combination of bands that contribute most to the model's performance. Specifically, the CARS algorithm employs Monte Carlo sampling and an exponentially decaying function to iteratively update the sampling probability of each band. Bands that have a stronger relationship with the target variables are assigned higher probabilities and are more likely to be retained [25].  
CARS is essentially a feature selection method based on model performance. By evaluating a large number of randomly generated band combinations, CARS effectively removes redundant and irrelevant bands, improving the model's interpretability and prediction accuracy. In hyperspectral data analysis, the CARS algorithm typically uses cross-validation to assess the performance of different band combinations and ultimately selects the subset of bands that minimize the cross-validation root mean square error (RMSEcv). The implementation of the CARS method can be described as follows.  
Use Monte Carlo sampling to randomly select a fixed proportion of samples from the dataset [26]. Establish a PLSR model using these samples. The PLSR model can be ex-pressed as follows:  
\[y=bX+e \tag{4}\]  
where  
\(y\) is the response variable vector;  
\(X\) is the predictor variable matrix;  
\(b={\big{[}b_{1},b_{2},\ldots,b_{n}\big{]}^{T}}\) is the regression coefficient vector;  
\(e\) is the error term vector.  
Evaluate the importance of each predictor variable (or spectral band) using the absolute values of the regression coefficients \(|b_{i}|\).  
Then, define a weight \(W_{i}\) for each variable to quantify its relative importance:  
\[W_{i}=\frac{|b_{i}|}{\sum_{j=1}^{n}{|b_{j}|}} \tag{5}\]  
Assign a weight of \(W_{i}=0\) to variables removed by the CARS algorithm.  
Iteratively eliminate variables with low weights using two methods. First, the Exponentially Decreasing Function (EDF) determines the number of variables to retain in each iteration. Then, Adaptive Reweighted Sampling (ARS) further refines the variable selection based on the weight.  
Repeat the upper steps for N iterations. Finally, select the subset of variables that yields the lowest Root Mean Square Error of Cross-Validation RMSE as the optimal combination.  
Since Monte Carlo sampling randomly selects a fixed proportion of samples in each iteration, rather than using all samples, the selected band combination has better adaptability. The EDF helps to quickly eliminate bands with low importance early in the iteration process, significantly reducing the computational burden. After multiple experiments, we set the number of Monte Carlo sampling iterations to 50, with 90% of samples used in each iteration.
## 2 Polar Subdivision
### Subdivision in circular layers  
To merge the refined polar structures and adjacent normal quadrilateral meshes subdivided by Catmull-Clark subdivision smoothly, we need to subdivide the polar structures in circular layers too. The refine ment rules in circular layers for polar subdivision can be described as:  
\[v^{\prime\prime}_{i,j}=\frac{1}{8}v^{\prime}_{i,j-1}+\frac{6}{8}v^{\prime}_{i,j}+ \frac{1}{8}v^{\prime}_{i,j+1},\hskip 28.452756 pte^{\prime\prime}_{i,j}=\frac{1}{2}(v^ {\prime}_{i,j}+v^{\prime}_{i,j+1}). \tag{4}\]  
The subdivision in circular layers is performed independently after or before subdivision in radial layers. The transformed refinement rules in circular layers:  
\[e^{\prime\prime}_{i,j}=e^{\prime}_{i,j}+\frac{1}{2}(v^{\prime}_{i,j}+v^{\prime }_{i,j+1}),\hskip 28.452756 ptv^{\prime\prime}_{i,j}=\frac{1}{4}(e^{\prime \prime}_{i,j-1}+2v^{\prime}_{i,j}+e^{\prime\prime}_{i,j}). \tag{5}\]  
We can verify that the results of Equation (5) are equal to the original rules, if \(e^{\prime}_{i,j}\) in expression (5) are initialized as zero. Knowing \(e^{\prime\prime}_{j}\), \(v^{\prime\prime}_{i,j}\), which are the points of mesh in higher resolution, we can get \(e^{\prime}_{j}\), \(v^{\prime}_{i,j}\) in expression (5) by inverting the subdivision process in circular layers:  
\[v^{\prime}_{i,j}=\frac{1}{2}(4v^{\prime\prime}_{i,j}-e^{\prime\prime}_{i,j-1} -e^{\prime\prime}_{i,j}),\hskip 28.452756 pte^{\prime}_{i,j}=e^{\prime\prime}_{ i,j}-\frac{1}{2}(v^{\prime}_{i,j}+v^{\prime}_{i,j+1}). \tag{6}\]  
Each sub-step of Equations (5 and 6) must be performed for all points of each type over the entire mesh before the next sub-step. The lifted polar subdivision and its inverted version have defined a lazy wavelet synthesis and analysis framework. The basis functions corresponding to \(e_{i,j}\) (denoted as lazy wavelet \(\psi\)), \(e_{j}\) (denoted as lazy wavelet \(\psi_{j}\)), \(v\) (denoted as scaling function \(\phi\)) have defined by the subdivision in radial layers. The basis functions corresponding to \(e^{\prime}_{i,j}\) (denoted as lazy wavelet \(\psi_{c}\)), \(v^{\prime}_{i,j}\) (denoted as lazy wavelet \(\phi_{c}\)) have been defined by the subdivision in circular layers.
# Unsupervised Text Feature Selection Using Memetic Dichotomous Differential Evolution
## 2 Preliminaries
### Document Pre-Processing
#### 2.4.2 Dichotomous Crossover  
The dichotomous crossover operation is utilized for producing the trial vector \(Ui\), by combining parts of both mutant vector \(V_{i,}\) and the target vector \(X_{i,}\) Figure 2 shows an illustrative example of the way that the dichotomous crossover is working. The trial vector, is produced by applying the following equations:  
\[u_{i,G}=\left\{\begin{array}{cc}v_{i,j,G}&if\ rand_{j}\leq Cr\\ x_{i,j,G}&otherwise\end{array}\right. \tag{8}\]  
\[Cr_{j}=\left\{\begin{array}{cc}Cr_{1}&if\ (x_{r1,G}\oplus x_{r2,G})==0\\ Cr_{2}&if\ (x_{r1,G}\oplus x_{r2,G})==1\end{array}\right. \tag{9}\]  
There is a similarity between both mechanisms of the standard DE crossover and the dichotomous one. However, the difference between them is that the dichotomous crossover has two crossover probabilities, while only one crossover probability is used with the standard DE. Considering the \(j\)th bit of a trial vector, if there are common bits between \(x_{r1,j,G}\), and \(x_{r2,j,G}\), then Cr\({}_{j}\) and Cr\({}_{1}\) are equal. On the other hand, if there are different bits between \(x_{r1,j,G}\) and \(x_{r2,j,G}\), then Cr\({}_{j}\) and Cr\({}_{2}\) are equal. As shown in Figure 2, the bits of the trial vector resulted from target vector \(x_{i,}\) and mutant vector V\({}_{i,}\) with difference probabilities of crossover. Figure 2 shows how to conduct the crossover for the first, second, third, and \(n\)th bit of \(U_{i,}\) depend on Cr\({}_{1}\), Cr\({}_{2}\), Cr\({}_{1}\), and Cr\({}_{2}\), respectively. In Algorithm 1, the above steps of the proposed dichotomous method used for the unsupervised feature selection is shown:  
```
Input: Population size \(N\), Crossover probability \(Cr\); objective function \(MAD\) Initialize the population \(P_{0}\) by random of \(N\) candidate solutions for\(i=1\) to \(N\) if\(X_{i,}0\) is an infeasible individual then  Execute Algorithm 1 for ratio-greedy repair endif  Evaluate the objective function value (\(Xi\),0) endfor \(FEs\) = NP while\(FEs\) < Max\(FEs\)do for\(i=1\): NPdo  Randomly select two individuals \(xr1\), and \(xr2\), from population \(PG\)  Execute the dichotomous mutation to generate a mutate individual \(Vi\),  Execute the dichotomous crossover to generate a trial individual \(Ui\), if\(Ui,G\) is an infeasible individual then  Execute Algorithm 1 for ratio-greedy repair endif  Evaluate the objective function value (\(Ui\),) if\(f(Ui,G)\) > \(f(Xi\),G)then \(Xi\), = \(Ui\),G endif endfor \(FEs\) = \(FEs\) + NP endwhile Output: Optimal individual with the maximum profit value
```  
**Algorithm 1** Dichotomous DE logical steps
## 3 Link Level Parameters and Channel Coefficient Generation
### Small Scale Parameters
#### 3.3.1 Cluster Delays  
The first SSP is the cluster delay. We need to generate with exponential delay distribution the propagation delays for each cluster as follows  
\[\tau_{n}^{\prime}=-r_{\tau}DS\ln(X_{n}) \tag{38}\]  
where \(r_{\tau}\) is the delay scaling, \(DS\) is the delay spread of the link and \(X_{n}\sim U(0,1)\) where \(n\) is the cluster index. For the final delay set, we need to normalize and sort the delay vector  
\[\tau_{n}=\text{sort}(\tau_{n}^{\prime}-\min(\tau_{n}^{\prime})). \tag{39}\]In case of \(LOS\) condition, additional scaling of delays is needed to compensate the effect of \(LOS\) peak addition to the delay spread. To scale the delay set, we use a constant \(C_{\tau}\): \(\tau_{n}^{LOS}=\tau_{n}/C_{\tau}\) where \(C_{\tau}=0.7705-0.0433K_{r}+0.0002K_{r}^{2}+0.000017K_{r}^{3}\) and \(K_{r}\) is the Ricean \(K\)-factor in dB.
# U-SSD: Improved SSD Based on U-Net Architecture for End-to-End Table Detection in Document Images
## 4 Experiment
### Evaluation
#### 4.3.2 Evaluation on Open Dataset
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multirow{2}{*}{**Model**} & \multicolumn{4}{c}{**IoU**} & \multirow{2}{*}{**Avg.**} \\ \cline{2-7}  & & **0.5** & **0.6** & **0.7** & **0.8** & **0.9** & \\ \hline \multirow{5}{*}{**Ours**} & Faster R-CNN + VGG & 0.83 & 0.82 & 0.81 & 0.73 & 0.54 & 0.75 \\  & Faster R-CNN + ResNet50 & 0.76 & 0.76 & 0.74 & 0.63 & 0.50 & 0.68 \\  & YOLOv3 & 0.95 & 0.95 & 0.95 & 0.86 & 0.50 & 0.84 \\  & YOLOv3 + U-Net & 0.96 & 0.96 & 0.94 & 0.87 & 0.57 & 0.86 \\  & SSD & 0.95 & 0.95 & 0.95 & 0.93 & 0.72 & 0.90 \\  & U-SSD & 0.95 & 0.95 & 0.95 & 0.94 & 0.89 & 0.94 \\ \hline \multirow{5}{*}{**Github**} & Faster R-CNN + VGG & 0.88 & 0.87 & 0.85 & 0.80 & 0.38 & 0.76 \\  & Faster R-CNN + ResNet50 & 0.82 & 0.81 & 0.80 & 0.74 & 0.32 & 0.70 \\  & YOLOv3 & 0.92 & 0.91 & 0.86 & 0.70 & 0.26 & 0.73 \\  & YOLOv3 + U-Net & 0.90 & 0.89 & 0.81 & 0.55 & 0.18 & 0.67 \\  & SSD & 0.89 & 0.89 & 0.88 & 0.80 & 0.31 & 0.75 \\  & U-SSD & 0.87 & 0.85 & 0.82 & 0.78 & 0.44 & 0.75 \\ \hline \multirow{5}{*}{**TableBank**} & Faster R-CNN + VGG & 0.96 & 0.96 & 0.96 & 0.92 & 0.50 & 0.86 \\  & Faster R-CNN + ResNet50 & 0.93 & 0.93 & 0.96 & 0.80 & 0.49 & 0.82 \\  & YOLOv3 & 0.97 & 0.97 & 0.90 & 0.79 & 0.41 & 0.81 \\  & YOLOv3 + U-Net & 0.96 & 0.94 & 0.89 & 0.66 & 0.33 & 0.76 \\  & SSD & 0.92 & 0.90 & 0.90 & 0.83 & 0.52 & 0.81 \\  & U-SSD & 0.93 & 0.93 & 0.92 & 0.86 & 0.60 & 0.85 \\ \hline \multirow{5}{*}{**ICDAR13**} & Faster R-CNN + VGG & 0.89 & 0.88 & 0.84 & 0.79 & 0.41 & 0.76 \\  & Faster R-CNN + ResNet50 & 0.85 & 0.84 & 0.80 & 0.75 & 0.43 & 0.73 \\  & YOLOv3 & 0.88 & 0.86 & 0.77 & 0.63 & 0.29 & 0.69 \\  & YOLOv3 + U-Net & 0.81 & 0.79 & 0.74 & 0.53 & 0.16 & 0.61 \\  & SSD & 0.89 & 0.88 & 0.87 & 0.79 & 0.37 & 0.76 \\  & U-SSD & 0.82 & 0.78 & 0.73 & 0.67 & 0.4 & 0.68 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The F1-scores from IoU = 0.5 to 0.9 on open dataset.
# Real-Time Monitoring of Wind Turbine Bearing Using Simple Neural Network on Raspberry Pi
## 4 Discussion  
Achieving a practical model requires a small model, with a high accuracy that can function in real time.
## 2 Materials and Methods  
### Background  
In the context of dimensionality reduction, given a dataset of \(n\) d-dimensional data points denoted as \(X=[x^{1},\ldots,x^{n}]\in\mathbb{R}^{d\times n}\) and a new target dimension \(t\) (\(t\ll d\)), the objective is to find a mapping function \(\phi\) that transforms the d-dimensional data points into a set of \(n\) t-dimensional points denoted as \(Y=[y^{1},\ldots,y^{n}]\in\mathbb{R}^{t\times n}\). The goal of this mapping is to minimize a specific distance function.  
Typically, the projection function aims to approximate the pairwise distance relationships between data points in X with those in Y. In other words, data points that are close to each other in X should also be close to each other in Y, and those that are distant in X should map to distant points in Y. The mapping function \(\phi\) relies on distance functions in both the high-dimensional space, denoted as \(\delta_{d}\), and the reduced space, denoted as \(\delta_{t}\). Frequently, the Euclidean distance \(L_{2}(p,p^{\prime})=\sum_{1\leq i\leq q}\sqrt{(p_{i}-p^{\prime}_{i})^{2}}\) is employed as the distance measure in both spaces. However, depending on the domain of the data, the distance (or dissimilarity) measure required may differ from the Euclidean distance. For instance, in the case of word embeddings, cosine similarity is commonly used as the distance measure.  
Indeed, when reducing the dimensionality of a dataset, not all distance properties can be fully preserved, as the reduced space is smaller than the original one. While preserving neighborhoods is a common way to evaluate the quality of a dimensionality reduction projection, there are other relevant aspects that should be considered. Some of these aspects include computational scalability, whether the transformation is deterministic or stochastic, and the preservation of certain geometric or topological properties of the data. For instance, preserving pairwise distances and neighborhood structures is crucial for many applications, but other factors like computational efficiency, interpretability of the results, and robustness to noise may also be important considerations.  
Our approach to evaluating the results of our algorithm is based on evaluating whether clustering improves after applying the vector modifications. As demonstrated in previous research, clustering can be improved using dimensionality reduction (DR) algorithms on the original data [29; 30]. However, our novel contribution is a new strategy for manipulating the original data before the application of DR. We demonstrate that our method improves clustering and thus provide valuable evidence of the utility and efficacy of our approach for data visualization.  
Our contributions are threefold:  
* A new method for high-dimensional vector manipulation that improves a wide range of DR algorithms.
* A validation study that proves that our technique enhances the results in many scenarios.  
* Data visualization examples that use document embeddings and provide evidence that the technique also works with other kinds of high-dimensional data.