## 3 Validation
### Experiment Setup  
The majority of dimensionality reduction (DR) algorithms demonstrate a nondeterministic behavior, primarily due to certain steps, such as initialization, which may utilize random values to begin the selection of elements. Additionally, algorithms like tSNE and UMAP are highly reliant on their hyperparameters [48]. For this study, we intentionally refrained from hyperparameter tuning to maintain fairness and avoid bias in the results. Fine-tuning the hyperparameters separately for both the original and modified datasets could potentially skew the evaluation, and we wanted to ensure a fair comparison between the two. To address this challenge and guarantee the robustness of our approach, we executed each DR algorithm on a five-time basis independently. This allowed us to obtain multiple sets \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Dataset** & **Dimensions** & **Samples** & **Classes** \\ \hline
20 NG & 99 & 18,844 & 20 \\ Cifar10 & 1024 & 3250 & 10 \\ Coil20 & 400 & 1440 & 20 \\ Fashion-MNIST & 784 & 10,000 & 10 \\ FlickMaterial10 & 1534 & 997 & 10 \\ Har & 561 & 735 & 6 \\ MNIST & 784 & 70,000 & 10 \\ Sentiment & 200 & 2748 & 2 \\ Spheres & 101 & 10,000 & 11 \\ Svhn & 1024 & 732 & 9 \\ USPS & 255 & 9298 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Features of the different datasets. of projections for each dataset, thereby minimizing the impact of the nondeterministic behavior and the influence of the default hyperparameter values. For the clustering task, we initially opted (later, we compared with other methods) for a linear Support Vector Machine (SVM) implementation by [PERSON] et al. [49] with the'multiclass' parameter. The SVM algorithm was executed for 1000 iterations and evaluated using two themes: 'calibrated' and 'not-calibrated,' with a tenfold cross-validation approach. During each iteration, the SVM classifiers were trained on 80% of the data, while the remaining 20% constituted the testing data. To address the nondeterministic nature of the outcomes resulting from the inclusion of random variables in the execution of the SVM algorithm, we conducted five repetitions of each experiment for each data sample. The resulting accuracy values were then averaged to ensure the reliability and consistency of the clustering performance evaluation. By using multiple repetitions, we aim to obtain robust and stable performance metrics for the SVM-based clustering algorithm across different datasets and DR techniques. To sum up, the process works as follows:  
* For each dataset \(d\), a transformation is applied using our algorithm, giving the transformed dataset \(d_{t}\). * We project both \(d\) and \(d_{t}\) using PaCMAP, tSNE, trimap, and UMAP. * Each projected set is evaluated five times with the linear SVM method, and the results are averaged. This procedure was repeated five times, and the results are the average of the five runs. The obtained results for the PaCMAP algorithm are presented in Table 2. It is evident that in the majority of cases, the accuracy improves when our method is applied to the various models and the data is subsequently reduced to two dimensions using PaCMAP. Specifically, nine out of the eleven models exhibited an increase in their clustering accuracy. The outcomes for the other dimensionality reduction (DR) algorithms exhibit similar trends. For the tSNE algorithm, we observe improvements in 7 out of the 11 models, as depicted in Table 3. The trimap algorithm does not work for the 20 NG dataset (an error occurred that we did not further investigate ). Nonetheless, the majority of the models (8 out of 10) also improve, with a behavior similar to that of PaCMAP, as indicated in Table 4. Similarly, the UMAP yields comparable results, with 8 out of the 11 models experiencing improved clustering accuracy, as shown in Table 5.  
Notably, the PaCMAP algorithm, as demonstrated in [1], is less sensitive to hyperparameters.