Bipolar Metric Spaces for Geometric Regularization in Distance-Based Learning

Distance-based machine learning algorithms are widely used due to their simplicity and flexibility. However, they often suffer from sensitivity to noise, high computational cost, and instability in high-dimensional and complex data spaces. To address these limitations, this study introduces a novel geometric learning framework based on Bipolar Metric Spaces (BMSs). The proposed framework extends classical metric structures by incorporating a quadrilateral inequality constraint, enabling the modeling of both similarity and contrast relationships across data points. Within this framework, geometric regularization is achieved through a structural filter mechanism that filters out structurally inconsistent assignments and stabilizes decision boundaries. Importantly, the proposed approach is not limited to specific algorithms and can be integrated into a broad class of distance-based learning methods. To demonstrate its effectiveness, the framework is instantiated on K-Nearest Neighbor (KNN) and K-means algorithms, resulting in Bipolar K-Nearest Neighbor (BP-KNN) and Bipolar K-means (BP-K-means). Theoretical analysis shows that the proposed framework improves Lipschitz stability, reduces variance, and enhances generalization performance. The models were tested on six different datasets, including benchmark datasets (Iris, Wine, and Breast Cancer) and real-world datasets (Abalone, CDC Diabetes, and Malicious Website Features (MWS)). Theoretical complexity analysis, supported by Floating Point Operation (FLOP) quantification, reveals that the bipolar approach offers a more stable decision boundary and significantly faster convergence, corroborating the observed efficiency improvements. Notably, on the large-scale MWS dataset, BP-KNN achieved an 11.6-fold increase in inference speed and a 75% reduction in RAM usage compared with the standard KNN. These results demonstrate that BMS provides a unified and scalable geometric regularization framework for distance-based learning, offering a principled solution to the trade-off between accuracy, robustness, and computational efficiency.