Structure-Preserving Clustering in Encrypted Feature Spaces

Abstract

Complex single valued neutrosophic sets (CSVNSs) extend conventional neutrosophic sets to complex valued membership functions, providing a flexible mathematical model of uncertainty, indeterminacy, and inconsistency. Since each component of a universe is represented by truth, indeterminacy, and falsity membership functions with values in the unit 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, and absolute set, are formally expressed and investigated. Information that exhibits phase-like or periodic behavior that can be described in the complex domain, in addition to being uncertain and incomplete, can be effectively represented by complex single valued neutrosophic sets (CSVNSs). In many real-world decision-making and pattern-recognition tasks, the evidence contains simultaneous degrees of truth, indeterminacy, and falsity. Additionally, classical similarity measures can become unstable when there is a preponderance of magnitude differences, scale dependence, and noise in the data. In order to assess the similarity between two CSVNS vectors, this study proposes a cotangent similarity assessment of Complex Single Valued Neutrosophic Sets (Cotangent CSVNS), which primarily relies on directional agreement and reduces the impact of raw distance. This study also presents privacy-sensitive clustering analysis using encrypted Iris data, both with and without feature normalization. Principal Component Analysis (PCA) is used to assess the quality of clustering using two-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, Encrypted K-Means and Encrypted K-Means++ are used to identify three clusters on the encrypted feature space. The encrypted data does not eliminate the data’s grouping tendencies, as evidenced by the two-dimensional PCA plots that show clusters that can be recognized as clearly distinguishable regions where the centroid values are well defined. Cluster boundaries are more balanced and smaller when normalization is performed before encryption since all feature magnitudes are scaled identically, increasing the reliability of distance separation. Additionally, the 3D PCA displays show that intra-cluster coherence and inter-cluster geometry are maintained by maintaining cluster separation in a more detailed reduced space with centroid separation and no group overlap. Centroid initialization, which selects spatially diverse starting points to improve convergence stability and reduce the likelihood of suboptimal partitions in encrypted applications, is improved by K-Means++. When the individual findings are combined, it is shown that unsupervised learning can be performed on encrypted and encrypted-normalized data while still producing meaningful structural patterns. This enables it to perform secure clustering, topology preservation, and analytics that respect privacy without revealing any underlying sensitive values.

keywords: Complex single-valued neutrosophic sets, Cotangent-based similarity measure, PCA-based visualization and topology preservation, Privacy-preserving encrypted K-Means clustering