Quadri-Partition Neutrosophic Soft Topology and Dimensionality Reduction, Clustering, and Signal Recovery in High-Dimensional Spaces

Quadri-partition neutrosophic soft locally compact spaces (QPNSLCS) of quadri-partition neutrosophic soft topological spaces (QPNSTS) are the concept introduced in this research. The theoretical foundation for the treatment of uncertainty in complex topological structures is strengthened by the fact that local compactness, particularly when combined with the Hausdorff condition, establishes the existence of compact neighborhoods and the compactness of subspaces. Parallel coordinate plots facilitate comparisons across multiple variables, PCA displays patterns of variance-based groupings, t-SNE in 2D and 3D displays patterns of similarity, and Figures 5.1 to 5.8 demonstrate various dimensionality reduction and clustering techniques to analyze high-dimensional data to support this theoretical discussion. The optimal number of clusters is estimated by the Elbow Method at K = 2, and comparisons between t-SNE and UMAP show that local and global structure preservations differ. Additionally, the purification of mixed signals with slight distortions is demonstrated using Independent Component Analysis (ICA). Together, these findings improve clustering mistakes, show structure patterns, reconstruct latent information, and offer a theoretical and practical knowledge of complicated data processing.