## 3 Validation
### Experiment Setup
While we anticipate that better results might be obtained through proper hyperparameter fine-tuning for the other methods, we deliberately kept the experiments as neutral as possible to avoid unintended bias. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Dataset** & **Original + PaCMAP (SVM)** & **Improved + PaCMAP (SVM)** \\ \hline
20 NG & 0.5156 & **0.8605** \\ Cifar10 & 0.2029 & **0.2252** \\ Coil20 & 0.8278 & **0.8639** \\ Fashion-MNIST & 0.7232 & **0.7424** \\ FlickMaterial10 & 0.5433 & **0.6047** \\ Har & 0.7285 & **0.7982** \\ MNIST & **0.9733** & 0.9416 \\ Sentiment & 0.6177 & **0.7989** \\ Spheres & 0.6653 & **0.9755** \\ Svhn & 0.1855 & **0.1955** \\ USPS & **0.9476** & 0.9401 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Impact of the vector manipulation algorithm on the PaCMAP DR method for clustering using SVM. The center column displays the accuracy of the clustering algorithm when no vector manipulation is applied to the data, while the rightmost column shows the accuracy when our manipulation approach is applied. Notably, in the majority of the cases (9 out of 11), the accuracy improves. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Dataset** & **Original + TIMAP (SVM)** & **Improved + TIMAP (SVM)** \\ \hline
20 NG & - & - \\ Cifar10 & 0.1914 & **0.2273** \\ Coil20 & 0.7977 & **0.8236** \\ Fashion-MNIST & 0.7161 & **0.7251** \\ FlickMaterial10 & 0.4780 & **0.6100** \\ Har & 0.6668 & **0.7620** \\ MNIST & **0.9636** & 0.8706 \\ Sentiment & 0.4812 & **0.9522** \\ Spheres & 0.7686 & **0.9757** \\ Svhn & 0.1891 & **0.1964** \\ USPS & **0.9387** & 0.9237 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Accuracy obtained using trimap dimensionality reduction. The center column shows the results without our method, while the rightmost use our algorithm. trimap breaks for the 20 NG model. In this case, we obtain an improvement in 8 out of the 10 models. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Dataset** & **Original + tSNE (SVM)** & **Improved + tSNE (SVM)** \\ \hline
20 NG & 0.4646 & **0.7854** \\ Cifar10 & 0.2031 & **0.2306** \\ Coil20 & **0.8319** & 0.8185 \\ Fashion-MNIST & 0.7222 & **0.7373** \\ FlickMaterial10 & 0.5633 & **0.6207** \\ Har & **0.8588** & 0.8317 \\ MNIST & **0.9666** & 0.9174 \\ Sentiment & 0.5852 & **0.8252** \\ Spheres & 0.7626 & **0.8933** \\ Svhn & 0.1909 & **0.1982** \\ USPS & **0.9528** & 0.9333 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: tSNE accuracy without (center) and with (right) the algorithm. In this case, 7 out of the 11 models improve the result. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Dataset** & **Original + tNMAP (SVM)** & **Improved + UMAP (SVM)** \\ \hline
20 NG & 0.4859 & **0.8076** \\ Cifar10 & 0.2062 & **0.2195** \\ Coil20 & 0.7894 & **0.8634** \\ Fashion-MNIST & 0.7247 & **0.7343** \\ FlickMaterial10 & 0.5833 & **0.6900** \\ Har & **0.8235** & 0.8054 \\ MNIST & **0.9650** & 0.9243 \\ Sentiment & 0.5927 & **0.6327** \\ Spheres & 0.5213 & **0.8933** \\ Svhn & 0.1955 & **0.2045** \\ USPS & **0.9520** & 0.9380 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Accuracy obtained using UMAP. The center column shows the results without our method, while the rightmost use our algorithm. Here, we obtained an improvement in 8 out of the 11 models. Figure 3: Evaluation of trimap clustering using KNN, DT, MLP, and XGBoost. As in the previous case, the results are analogous to the ones obtained with SVM. Figure 2: Comparing other clustering algorithms for the PaCMAP DR algorithm. The results are equivalent to the ones obtained with SVM. Figure 4: Results obtained with the additional clustering algorithms for tSNE. Figure 5: Analysis of the UMAP DR data. The additional clustering algorithms also work similarly to the SVM. To provide further insights into the performance of the algorithms, we compiled a summary of the percentages of the improvement in Table 6. The table reveals that our vector manipulation strategy improves the clustering accuracy for the majority of datasets.