Integrating Lipschitz Extensions and Probabilistic Modelling for Metric Space Classification

Lipschitz-based classification provides a flexible framework for general metric spaces, naturally adapting to complex data structures without assuming linearity. However, direct applications of classical extensions often yield decision boundaries equivalent to the 1-Nearest Neighbour classifier, leading to overfitting and sensitivity to noise. Addressing this limitation, this paper introduces a novel binary classification algorithm that integrates probabilistic kernel smoothing with explicit Lipschitz extensions. We approximate the conditional probability of class membership by extending smoothed labels through a family of bounded Lipschitz functions. Theoretically, we prove that while direct extensions of binary labels collapse to nearest-neighbour rules, our probabilistic approach guarantees controlled complexity and stability. Experimentally, evaluations on synthetic and real-world datasets demonstrate that this methodology generates smooth, interpretable decision boundaries resilient to outliers. The results confirm that combining kernel smoothing with adaptive Lipschitz extensions yields performance competitive with state-of-the-art methods while offering superior geometric interpretability.