esa-sceva/satcom-chunk-collection huggingface dataset:
# Validation of an RF Image System for Real-Time Tracking Neurosurgical Tools
## 3 Experimental Validation and Results
### Data Extraction and Results
The mean error (\(\overline{\Delta}\)) and the standard deviation (\(\sigma\)) for each coordinate (\(x\) and \(y\)) were independently considered in this analysis. Additionally, different ranges of positions are involved, depending on the different regions where the tool navigated: p3 to p8 includes the tool travelling through the antenna system inner space, the cranium boundary, and the cranium inner space, p5 to p8 includes the tool travelling through the cranium boundary and its inner space, and p6 to p8 includes only the tool travelling through the cranium inner space. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Position Range**} & \multicolumn{3}{c}{**DAS**} & \multicolumn{3}{c}{**DMAS**} & \multicolumn{3}{c}{**Robot**} \\ \cline{2-9}  & \(\overline{\Delta x}\) & \(\overline{\Delta y}\) & \(\sigma_{x}\) & \(\sigma_{y}\) & \(\overline{\Delta x}\) & \(\overline{\Delta y}\) & \(\sigma_{x}\) & \(\sigma_{y}\) \\ \hline p3 to p8 & 8.2593 & 11.5358 & 8.2550 & 11.0136 & 8.2721 & 12.8285 & 8.4759 & 11.5053 \\ p5 to p8 & 3.4465 & 5.3047 & 4.3779 & 6.8422 & 3.3456 & 6.2968 & 4.6237 & 7.0526 \\ p6 to p8 & 1.4368 & 2.0563 & 2.1249 & 2.6289 & 1.2575 & 3.0183 & 2.4299 & 3.1812 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Error analysis for both algorithms (data in mm). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Positions**} & \multicolumn{2}{c}{**DAS**} & \multicolumn{2}{c}{**DMAS**} & \multicolumn{2}{c}{**Robot**} \\ \cline{2-9}  & \(x\) & \(y\) & \(x\) & \(y\) & \(x\) & \(y\) \\ \hline Tumor & \(-0.6342\) & 26.1538 & 0.0961 & 26.3117 & — & — \\ p2 & 80.6823 & 0.4284 & 111.0871 & \(-13.9562\) & 90.4000 & 96.2000 \\ p3 & 47.4907 & 47.0544 & 47.4907 & 44.3263 & 67.0000 & 70.8000 \\ p4 & 42.3399 & 37.4496 & 41.8595 & 36.3899 &Finally, the influence of the luminance threshold was analyzed. Given the agreement between the detected coordinates with both algorithms at p8 and the reference coordinates from the robot arm, this position was taken as general reference. The detected coordinates were recalculated from the saved measurements with DAS and DMAS algorithms using different luminance thresholds. For each newly recalculated pair of coordinates (associated to a certain luminance threshold), the detection error (\(DE\)) was computed as the Euclidean distance to the reference coordinates:  
\[DE=\sqrt{\left|x_{d}-x_{r}\right|^{2}+\left|y_{d}-y_{r}\right|^{2}} \tag{6}\]  
where \(x_{d}\) and \(y_{d}\) are the detected coordinates, whilst \(x_{r}\) and \(y_{r}\) are the reference coordinates. The resulting evolution of the \(DE\) depending on the luminance threshold for DAS and DMAS algorithms at p8 is plotted in Figure 18. It should be noted that this plot was made from only one measurement (p8), which was analyzed many times with different luminance thresholds. It therefore gives information about the digital error when computing the coordinates as a function of this threshold.
## 5 Description of the Laboratory Test and the Numerical Simulations
### Nonlinear Numerical Model  
The laboratory tests are simulated with a nonlinear f.e.m. model created with DIANA FEA software version 10.4 [34; 35]. 3D solid (brick) elements with 20 nodes (Figure 12a) are used for modelling the concrete bodies. Reinforcements are modelled with bar-embedded elements (Figure 12b) in the hypothesis of a full bond between concrete and steel.  
Brick elements of 30 mm side are used for the concrete parts not in contact with the jacketing plates (Figure 13a), whereas the bricks in contact with the steel plates have a maximum side size of 15 mm (Figure 13b). The mesh dimensions are chosen in order to obtain numerical results not dependent on the mesh size. A Coulomb attritive interface is modeled between steel jacketing plates and concrete. The rebar layout is shown in Figure 14. The supports and loading plates are modeled by means of elastic steel elements to ensure adequate stress distribution and realistic stress concentration in the concrete elements. The interface between support plates and concrete is also modeled with a Coulomb attritive surface. The support plates have plan dimensions of 20 \(\times\) 20 cm and a thickness of 4 cm. The load plate has plan dimensions of 20 \(\times\) 40 cm and a thickness of 4 cm. The support plates are constrained in their midline with a hinge (on one side) and a roller (on the other side).  
Figure 11: Stress-strain relationship of reinforcement specimens from laboratory tests.  
Figure 12: Elements used for nonlinear 3D modelling [35]. (**a**) A 20 nodes brick element. (**b**) Embedded bar element.  
A Smeared Cracking model is used for concrete [36] with a constitutive law based on the total strain crack model, which in DIANA FEA grounds on the Modified Compression Field Theory [37] extending it to three-dimensional problems following [PERSON] & Vecchio [38]. A linear-ultimate crack strain model is used in tension, with a linear tension softening branch after reaching peak tensile strength (Figure 15a). The stress-strain relationship for concrete in compression is based on EN 1992-1-1 [30] (Figure 15b). The reduction of compressive strength due to lateral cracking is based on the model proposed in [39]. A confinement model based on the proposal by [PERSON] & Collins [40] is adopted, and a constant shear retention factor \(\beta=0.2\) is used to describe shear stiffness after cracking.  
Figure 14: The rebars layout. **(a)** Beam with orthogonal reinforcement in the dapped end (**b**) beam with inclined reinforcement in the dapped end.  
Figure 15: Stress-strain relationship for concrete [41]. **(a)** Tension (**b**) Compression.  
Elastic-plastic stress-strain relationship with linear hardening is used for reinforcement. Yield strength is 500 MPa, whereas ultimate strength is 550 MPa. An ultimate strain of 10% is chosen.  
Jack load is applied with increments of 8 kN per step. The Newton-Raphson integration method is used for reaching convergence.
# CPF-UNet: A Dual-Path U-Net Structure for Semantic Segmentation of Panoramic Surround-View Images
## 2 Methods
### Network Architecture
#### 2.3.1 Main Branch  
The resulting feature maps are designated as the Semantic Branch, while the third layer of the decoder is used as the Detail Branch, both being directed into the FFM for the integration of features. Inside the FFM, a thorough interaction and fusion take place between the deep and refined shallow features. Upon entering the FFM, the Semantic Branch is split into two pathways, referred to as SemanticBranch_1 and SemanticBranch_2. Likewise, the Detail Branch is divided into DetailBranch_1 and DetailBranch_2. Depthwise convolutions are individually applied to DetailBranch_1 and SemanticBranch_1, with their respective input feature maps denoted as \(X_{Detail}\in R^{H\times W\times C}\) and \(X_{Semantic}\in R^{H/4\times W/4\times C}\) here, focusing on \(X_{Semantic}\) as an example:  
\[X_{DW}^{(c)}[m,n]=\sum_{i=1}^{K}\sum_{j=1}^{K}W_{ij}^{(c)}\cdot X_{Semantic}^{( c)}[m+i,n+j] \tag{5}\]  
here, \(X_{DW}^{(c)}[m,n]\) represents the value of the \(c^{th}\) channel at position \((m,n)\) in the output feature map, \(W_{ij}^{(c)}\) denotes the element within the depthwise convolution kernel corresponding to channel \(c\), and \(X_{Semantic}^{(c)}[m+i,n+j]\) signifies the value of channel \(c\) in the input feature map at the relative position \((i,j)\). Subsequently, the feature maps resulting from the depthwise convolution undergo pointwise convolution to adjust the number of channels:  
\[P_{j}[m,n]=\sum_{i=1}^{inclannels}w^{\prime}{}_{ij}X_{DW}[m,n,i] \tag{6}\]  
Figure 3: Overall network architecture diagram.  
wherein, \(w^{\prime}{}_{ij}\) represents the weight parameter for the pointwise convolution, and \(P_{j}\) denotes the \(j^{th}\) channel of the output feature map. After obtaining \(P_{S}\) and \(P_{D}\) through depthwise separable convolution from the Semantic Branch and Detail Branch, respectively, both primary branches undergo batch normalization. Focusing on the Semantic Branch as an example:  
\[X_{BN\_S}=\gamma\frac{P_{S}-\mu}{\sqrt{\sigma^{2}+\epsilon}}+\beta \tag{7}\]  
a convolution with a \(1\times 1\) kernel:  
\[X_{Conv\_semantic}=\sum_{i=1}^{1}\sum_{j=1}^{1}W_{ij}\cdot X_{BN\_S}[i,j] \tag{8}\]  
following these operations, we obtain \(X_{Conv\_Semantic}\) and \(X_{Conv\_Detail}\), respectively. Subsequently, \(X_{Conv\_semantic}\) is passed through a sigmoid activation function:  
\[X_{Sigmoid\_Semantics}=\sigma(X_{Conv\_semantic}) \tag{9}\]  
where \(\sigma\) represents the sigmoid activation function, \(\mu\) denotes the mean of the batch, \(\sigma^{2}\) signifies the variance, \(\gamma\) is the scaling factor, and \(\beta\) represents the offset parameter.
# Autonomous Active Tag Using Energy Harvesting Strategies
## 2 Materials and Methods
### Power Module
#### 2.2.2 Solar Submodule  
This submodule harvests energy radiated from the sun using a solar panel. In the preliminary tests, two solar panels were tested but we selected the AM-5608 CAR, with an estimated operating time of 40 years. This was chosen following two constraints: to optimize the size of the tag developed (the selected panel has a size of 60.10 mm \(\times\) 41.30 mm \(\times\) 1.80 mm) and the voltage range of the supercapacitor for the sake of efficiency. The other panel used in the preliminary tests (MONOCRYSTL SOLAR CELL 313070004) has a size of 70 mm \(\times\) 55 mm \(\times\) 1.5 mm and a voltage range greater than the one needed.  
The output voltage of the solar panel is connected to a step-up DC/DC converter, which charges the supercapacitor and provides a constant charging voltage, as well as preventing damage to the supercapacitor from high fluctuations. Although this strategy causes energy losses due to the efficiency of the converters, this efficiency is considered enough for our application.  
The chosen converter is the LTC3105, which is used in energy harvesting applications. This converter meets the abovementioned specifications and, moreover, has good energy efficiency thanks to the Maximum Power Point Control (MPPC), which maximizes the energy that can be extracted from any source of power. The output voltage was set to 4.2 volts, which is the charging voltage of the supercapacitor.
## 4 Results and Discussion  
The performance of the dead reckoned INS (INS DR) and proposed NN approaches are analysed comparatively across several GPS outage simulated sequences, each of 10 s length. The positioning techniques are first analysed on a less challenging scenario involving vehicle travel on an approximate straight line on the motorway. Further analysis is then done on more challenging scenarios such as hard brake, roundabouts, quick changes in acceleration and sharp cornering and successive left and right turns using the performance metrics defined in Section 3.1.
# Triple-Band Notched Ultra-Wideband Microstrip MIMO Antenna with Bluetooth Band
## 4 Results and Discussion
### Diversity Analysis Characteristics (ECC, DG, and TARC) for Proposed UIWB-MIMO Antennas
\(S_{21}\) and \(S_{12}\) are the transmission coefficients between port 1 and port 2. The \(\theta\) is the phase value for the input signal. Figure 26 shows the TARC curves at different values of theta from 0\({}^{\circ}\) to 180\({}^{\circ}\). Figure 24: The simulated ECC versus frequency generated from far field. Finally, Table 3 shows a comparison of the proposed UWB-MIMO antenna with other UWB-MIMO antennas. Figure 26: TARC curves at different values of theta from 0\({}^{\circ}\) to 180\({}^{\circ}\). Figure 25: The simulated diversity gain versus frequency. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline
**Ref.** & **Size (mm\({}^{\circ}\))** & **Dileclecation** & **Frequency** & **Nonk Bands (GHz)** & **Basepoint** & **location (dB)** & **Fok Cain (dB)** & **Rad. Eff. (cc)** & **ECC** & **Ground PlanetGrid** & **Shapeness** \\  & **Constant** & **Range (GHz)** & **Target (GHz)** & **Support** & **Isoperimization (dB)** & **Fok Cain (dB)** & **Rad. Eff. (cc)** & **ECC** & **Completion** & **Edge-of** \\ \hline
[16] & 60\(\times\) 60 \(\times\) 1.6 & 4.4 & 2.73-01.68 & WLAN (S.36+0.40) & No & Below \(-15\) & 5.5 & N/A & -0.0015 & No CP Connected & No \\
[19] & 64.4 \(\times\) 68 \(\times\) 0.8 & 4.4 & 3-11 & WLAN (S.41) & No & Below \(-20\) & 5.5 & 80 & -0.005 & Connected & No \\
[20] & 78.8 \(\times\) 0.705 & 2.2 & 29-21.50 & WLAN (S.41) & No & Below \(-20\) & 5.3 & 90 & -0.117 & No CP Connected & No \\
[21] & 80.0 \(\times\) 1.52 & 3.5 & 3-18.15 & NoNoNo & Below \(-15\) & 6.6 & 96 & -0.012 & No CP Connected & No \\
[22] & 50.5 \(\times\) 1.52 & 4.5 & 3-12 & WLAN (S.48+0.53) & No & Below \(-17\) & 5.4 & 96 & -0.05 & No CP Connected & No \\
[23] & 80.0 \(\times\) 1.24 & 4.4 & 3-10 & WLAN (S.49) & No & Below \(-25\) & 5.8 & 90 & -0.05 & No CP Connected & No \\
[24] & 80.0 \(\times\) 1.24 & 4.4 & 3-10 & WLAN (S.49) & No & Below \(-25\) & 5.8 & 90 & -0.05 & CP Connected & No \\
[25] & 78.8 \(\times\) 1.5 & 4.5 & 2.35-16 & WLAN (S.49) & No & Below \(-29\) & 5.1 & 90 & -0.05 & Compared & No \\
[26] & 78.8 \(\times\) 1.5 & 4.5 & 2.35-16 & WLAN (S.49) & No & Below \(-29\) & 5.1 & 90 & -0.05 & Compared & No \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison with existing literature.
## 2 Materials and Methods
### Characterization Techniques  
The textural properties of MWCNTs, \(\gamma\)-Fe\({}_{2}\)O\({}_{3}\) and A/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\)/f-CNT were explored by adsorption-desorption of N\({}_{2}\) at 77 K in an ASAP 2020 system (Micromeritics, Norcross, GA, USA). Before the analysis, the samples were outgassed at 250 \({}^{\circ}\)C for 3 h. The specific surface area (S\({}_{\text{BET}}\), m\({}^{2}\) g\({}^{-1}\)) of the materials was evaluated by Brunauer, Emmett and Teller (B.E.T.) equation in a pressure range of P/P\({}^{0}\) = 0.15-0.35. In addition, the external surface area (S\({}_{\text{ext}}\), m\({}^{2}\) g\({}^{-1}\)) was determined according to _t-plot_ method and the total pore volume value (V\({}_{\text{Total}}\), cm\({}^{3}\) g\({}^{-1}\)) considered was at a relative pressure of P/P\({}^{0}\) = 0.99. FTIR spectra of \(\gamma\)-Fe\({}_{2}\)O\({}_{3}\), f-CNT, A, A/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\) and A/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\)/f-CNT before and after MB adsorption were obtained in a FTIR 8400S spectrometer (Shimadzu, Kyoto, Japan) in the range of 400-4000 cm\({}^{-1}\)using KBr pellet technique. The magnetic properties of \(\gamma\)-Fe\({}_{2}\)O\({}_{3}\), A/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\) and A/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\)/f-CNT were determined using a 7404-S vibrating sample magnetometer (Lake Shore, Princeton, Wi, USA) at room temperature. The crystalline phases in \(\gamma\)-Fe\({}_{2}\)O\({}_{3}\), f-CNT and A/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\)/f-CNT materials were analysed by X-ray diffraction (XRD) using an X'Pert PRO diffractometer (Malvern Panalytical Inc., Westborough, MA, USA) with Cu K\(\alpha\) radiation (45 kV, 40 mA) in the 20 range between 10\({}^{\circ}\) and 80\({}^{\circ}\) at a scanning rate of 2\({}^{\circ}\) min\({}^{-1}\). The spectra were analyzed by the Rietveld Refinement method using the Material Analysis Using Diffraction (MAUD) program and the average crystal size was calculated using the Debye-Scherrer equation. Thermogravimetric analysis (TGA/DTG) was conducted in an EXTAR 6000 thermal analyzer (Seiko, Mahwa, NJ, USA) under nitrogen atmosphere at a heating rate of 10 \({}^{\circ}\)C min\({}^{-1}\), from 30 to 900 \({}^{\circ}\)C. Thus, TGA/DTG analysis in air atmosphere (from 30 to 900 \({}^{\circ}\)C, 10 \({}^{\circ}\)C min\({}^{-1}\)) of \(\gamma\)-Fe\({}_{2}\)O\({}_{3}\) particles, f-CNT, alginate/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\) beads and A/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\)/f-CNT composite were accomplished. Finally, the surface morphology of the materials was studied by scanning electron microscopy (SEM) using a JSM 6400 microscope (JEOL, Peabody, MA, USA) equipped with a thermoionic cathode and a 25 kV detector. EDX analysis of A/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\)/f-CNT was accomplished. Additionally, the morphology of functionalized carbon nanotubes (f-CNT) was explored by transmission electronic microscopy (TEM) in a JEOL JEM 2100 microscope (200 kV, 0.25 nm of resolution).
## 2 Theoretical Background
### Equivalent Analytical Model for Calculating the URI  
Assuming that the LSF in structure is a strictly monotone function with respect to uncertain variables, URI can be estimated by the following crisp equivalent analytical model.  
**Theorem 3**.: _Consider the LSF \(g(\xi_{1},\xi_{2},\cdots,\xi_{n})\) of a structure system that contains independent uncertain variables \(\xi_{1},\xi_{2},\cdots,\xi_{n}\)with regular uncertainty distributions \(\Phi_{1},\ \Phi_{2},\cdots,\ \Phi_{n}\), respectively. If \(g(\xi_{1},\xi_{2},\cdots,\xi_{n})\) is continuous and strictly increasing for \(\xi_{1},\xi_{2},\cdots,\xi_{n}\) while strictly decreasing for \(\xi_{n+1},\xi_{n+2},\cdots,\xi_{n}\), then the URI is the root \(\alpha\) of the following equation:_  
\[f(\Phi_{1}^{-1}(1-\alpha),\ \Phi_{2}^{-1}(1-\alpha),\cdots,\ \[PERSON]_{m}^{-1}(1- \alpha),\ \[PERSON]_{m+1}^{-1}(\alpha),\[PERSON]_{m+2}^{-1}(\alpha),\cdots,\ \[PERSON]_{n}^{-1}(\alpha))=0 \tag{10}\]  
**Proof of Theorem 3.** According to Theorem 1, \(\xi=-g(\xi_{1},\xi_{2},\cdots,\xi_{n})\) is also an uncertain variable with a following inverse uncertainty distribution:  
\[\mathbb{F}_{-g(\mathbf{\xi})}^{-1}(\alpha)=f(\[PERSON]_{1}^{-1}(1-\alpha),\ \[PERSON]_{2}^{-1}(1-\alpha),\ \cdots,\ \Phi_{m}^{-1}(1-\alpha),\ \[PERSON]_{m+1}^{-1}(\alpha),\Phi_{m+2}^{-1}(\alpha),\cdots,\ \Phi_{n}^{-1}(\alpha)) \tag{11}\]  
Therefore, \(\mathrm{URI}=\mathcal{M}\{g(\mathbf{\xi})>0\}=\mathcal{M}\{-g(\mathbf{\xi})<0\}= \mathbb{F}_{-g(\mathbf{\xi})}(0)=\alpha\) is the root of the equation \(\mathbb{F}_{-g(\mathbf{\xi})}^{-1}(\alpha)=0\). The proof is completed. \(\square\)  
However, the monotonicity of LSF with respect to corresponding uncertain variables may not be determined, and Equation (10) sometimes has no solution. To overcome this problem, the decision-maker can estimate URI according to the US algorithm described in Section 3.2.
## 3 Methods
### Extraction of Human Pose Euler Angles
#### 3.2.3 Attitude Angle Calculation Algorithm Design  
A new method of human attitude angle calculation based on multiple attitude parameters compensating each other is proposed, which makes up for the redundant parameters and the singular values of trigonometric functions. The attitude angle calculation procedure begins with the initialization process and unit conversion of acceleration data (lines 1-2). Subsequently, Euler angles are computed based on the sampled frequency-processing sequence data, with compensation provided through gyroscope and accelerometer data (lines 3-12). Following this, the Euler angle vector is transformed into a rotation matrix, and from there into a quaternion and Rodrigues parameters (lines 13-15). Finally, the ultimate pose angles are calculated and displayed (lines 16-19).  
```
Input: Sample time series \(T\), Acceleration data \(A_{data}\) (\(a_{x}\), \(a_{y}\), \(a_{z}\)), Gyroscope data \(G_{data}\), Magnetometer data \(M_{data}\); Output: Attitude Angle \(attitude\); (1) Initialization \(A_{data}\), \(M_{data}\), Quaternion \(Q\) and Rodrigues Parameters \(r\); (2) Conversion unit of accelerometer data to the acceleration of gravity (3) for\(i\) = 0,0.04,0.08..T do: (4) Calculate roll angle \(\varnothing_{t}\) and pitch angle \(\theta_{t}\) (5) Compensate for \(M_{data}\) to correct Attitude Angle with magnetometer (6) Calculate yaw angle \(\varphi_{t}\) (7) Convert angles to radians (8) Update Euler Angle vector \(E_{angle}\) (9) Calculate the Gyroscope Euler Angle variation: (10) \(G_{change}=G_{data}\)\({}^{*}d_{t}\) (11) Compensate the Euler Angle \(C_{comp}\): \(C_{comp}=E_{angle}+G_{data}\); (13) Transform Euler Angle vector to rotation matrix (14) Convert Euler Angle vector to quaternion (15) Convert quaternion to Rodrigues Parameters (16) Calculate Attitude Angle \(attitude\) (17) Convert radians to angles (18) endfor (19) Output the final Attitude Angle \(attitude\)
```  
**Algorithm 1**The human attitude angle calculation method based on multiple attitude parameters
## 3 Error Analysis and Pointing Model  
### Sources of Pointing Error  
According to Figure 4, the pointing error sources of MPEOT can be categorized into four main groups [31]. Firstly, equipment-related pointing errors include (1) errors of the optical system, mainly caused by disturbance and inconsistency between the boresight of various optical components, (2) errors of electro-mechanical gimbals, which refer to geometric errors between axes, such as perpendicularity error, intersection error, misalignment  
Figure 3: Definition of pointing error.  
Figure 2: Target pointing process of EOT.  
error, and readout transducer error, and (3) errors of the servo control system, specifically the tracking error and stability error. Secondly, misalignment between the telescope base and the platform navigation system can be classified into two types: (1) orientation alignment error and horizontal leveling error during initial setup, and (2) angular vibration error during the operation. Next, the inaccuracy of the platform navigation system, notably the attitude measurement error of the inertial navigation system (INS), should be considered. Meanwhile, factors such as temperature, gravity, wind, air pressure, vibration, and shock can also have different effects on the above errors. To unify the coupling relationship between the static and dynamic errors of the system, this dissertation delineates several error coordinate frames in detail, derives their coordinate transformation matrices, and then establishes a telescope-pointing model.
# Well-Separated Pair Decompositions for High-Dimensional Datasets
## 4 Experiments
### Datasets  
We evaluated the computation of a WSPD on artificially generated and real high-dimensional datasets. Artificially generated datasets are drawn from the following:  
* A uniform distribution over the \([0,1]^{d}\) hypercube;
* A normal (Gaussian) distribution, with mean 0 and standard deviation 1;
* A Laplace or double exponential distribution, with the position of the distribution peak at 0 and the exponential decay at 1.  
For real-world datasets, I used two public scRNA-seq datasets downloaded from [17]. scRNA-seq data are used to assess which genes are turned on in a cell and in what amount. Therefore, scRNA-data are typically used in computational biology to determine transcriptional similarities and differences within a population of cells, allowing for a better understanding of the biology of a cell. We apply the standard preprocessing to scRNA-seq data (see [18; 19]): (a) compute the natural log-transformation of gene counts after adding a pseudo count of 1 and (b) select the top 2000 most variable genes followed by a dimensionality reduction to 100 principle components. We used two datasets of sizes \(n=4185\) and \(n=68,575\).  
One possible motivation for a WSPD on scRNA-seq data can be found in the work of [20]. Namely, the authors in [20] solve the marker gene selection problem by solving a linear program (LP). For a large amount of data, the LP cannot be efficiently solved in practice. Hence, the practical approach to that issue could be to solve an LP on a subset of constraints, then define a separation oracle that iteratively introduces new constraints to the LP, if the current solution is not feasible. Following the work of [20], the oracle could add the currently most violated constraints, i.e., pairs of points whose distance is less than a predefined constant value. Given the WSPD, one could speed up the oracle by using dumbbells as a substitute for the pairwise distances.