Structure-Preserving Clustering in Encrypted Feature Spaces

Complex single valued neutrosophic sets (CSVNSs) extend conventional neutrosophic sets to complex valuedmembership functions, providing a flexible mathematical model of uncertainty, indeterminacy, and inconsistency. Sinceeach component of a universe is represented by truth, indeterminacy, and falsity membership functions with values in theunit interval, this enables the expression of richer and more expressive information in discrete and continuous models.Basic set-theoretic concepts of CSVNSs, such as subset hood, equality, union, intersection, complement, null set, andabsolute set, are formally expressed and investigated. Information that exhibits phase-like or periodic behavior that canbe described in the complex domain, in addition to being uncertain and incomplete, can be effectively represented bycomplex single valued neutrosophic sets (CSVNSs). In many real-world decision-making and pattern-recognition tasks, theevidence contains simultaneous degrees of truth, indeterminacy, and falsity. Additionally, classical similarity measures canbecome unstable when there is a preponderance of magnitude differences, scale dependence, and noise in the data. In orderto assess the similarity between two CSVNS vectors, this study proposes a cotangent similarity assessment of ComplexSingle Valued Neutrosophic Sets (Cotangent CSVNS), which primarily relies on directional agreement and reduces theimpact of raw distance. This study also presents privacy-sensitive clustering analysis using encrypted Iris data, both withand without feature normalization. Principal Component Analysis (PCA) is used to assess the quality of clustering usingtwo-dimensional and three-dimensional visualization. While PCA is only used to map the encrypted representations to low-dimensional views summarizing the largest variance directions and making the cluster structure interpretable, EncryptedK-Means and Encrypted K-Means++ are used to identify three clusters on the encrypted feature space. The encrypted datadoes not eliminate the data’s grouping tendencies, as evidenced by the two-dimensional PCA plots that show clusters thatcan be recognized as clearly distinguishable regions where the centroid values are well defined. Cluster boundaries aremore balanced and smaller when normalization is performed before encryption since all feature magnitudes are scaledidentically, increasing the reliability of distance separation. Additionally, the 3D PCA displays show that intra-clustercoherence and inter-cluster geometry are maintained by maintaining cluster separation in a more detailed reduced spacewith centroid separation and no group overlap. Centroid initialization, which selects spatially diverse starting points toimprove convergence stability and reduce the likelihood of suboptimal partitions in encrypted applications, is improvedby K-Means++. When the individual findings are combined, it is shown that unsupervised learning can be performed onencrypted and encrypted-normalized data while still producing meaningful structural patterns. This enables it to performsecure clustering, topology preservation, and analytics that respect privacy without revealing any underlying sensitive values.