APG-AMS: Entropy-Guided Screening for Adaptive Metric Search in Distance-Based Learning

Distance-based learning algorithms depend critically on the choice of metric. Exhaustive search over candidate Minkowski norms may improve local neighborhood behavior, but it also increases computational cost through repeated distance evaluations. This paper introduces Arithmetic Power Geometry Adaptive Metric Selection (APG-AMS), an entropy-guided screening framework for reducing unnecessary adaptive metric search. The method is motivated by a local APG exponent-deformation result: near the Euclidean reference exponent p = 2, the first-order sensitivity of the normalized power-sum distribution is governed by the Shannon entropy of the squared-coordinate weight distribution. APG-AMS uses normalized APG entropy to identify potentially metric-sensitive query samples and applies full local p-search only when the entropy criterion predicts elevated metric sensitivity. Experiments were conducted on eight benchmark and controlled datasets using 10-fold stratified cross-validation. At threshold τ = 0.92, APG-AMS produced predictive accuracy statistically indistinguishable from both Euclidean k-nearest neighbors and full adaptive p-search. The primary evaluation used 10-fold stratified cross-validation, while the earlier 70/30 split experiment is reported only as an auxiliary threshold-ablation study. Across 80 paired folds, APG-AMS versus Euclidean k-NN yielded paired t-test p = 0.417 and Wilcoxon p = 0.289, while APG-AMS versus full p-search yielded paired t-test p = 0.864 and Wilcoxon p = 0.641. The results suggest that APG entropy can serve as a practical screening indicator for metric stability in the evaluated datasets. The proposed framework is not intended as a replacement for supervised metric-learning methods, but rather as a computational screening layer for adaptive metric search.