K-Nearest Neighbor Algorithm Based on the Framework of Ordered Pair of Normalized Real Numbers

The K-Nearest Neighbors (kNN) algorithm, a cornerstone of supervised learning, relies on similarity measures constrained by real-number-based distance metrics. A critical limitation of traditional kNN research lies in its confinement to the real-number domain, which inherently restricts its ability to model nonlinear feature interactions in high-dimensional data and amplifies sensitivity to feature redundancy and class imbalance. These limitations arise from the inherent linearity and unidimensional nature of real-number representations, which restrict their ability to model complex feature interdependencies. To transcend these limitations, this paper proposes OPNs-kNN, a novel framework grounded in Ordered Pairs of Normalized Real Numbers (OPNs). Departing from the conventional real-number paradigm, OPNs-kNN constructs feature pairs as multidimensional OPNs tuples and employs a generalized OPNs-valued metric to explicitly model nonlinear relationships, thereby addressing the inherent shortcomings of real-number-based kNN. Extensive experiments on nine UCI benchmark datasets (e.g., glass, wines, seeds) demonstrate that OPNs-kNN achieves statistically significant improvements in classification accuracy, precision, recall, and F1 score compared to traditional kNN and its enhanced variants. This work pioneers a non-real-number computational framework, proving that moving beyond real-number constraints enables more expressive representations of data relationships, opening new directions for designing robust machine learning models in complex domains.