From distance to gravitation: A new perspective for improving classification, clustering, and sampling

Distance-based learning methods—including K-means and support vector machines (SVMs)—rely heavily on Euclidean distance to measure similarity, but they overlook discriminative information in vector magnitudes (norms) by treating data points purely as geometric coordinates rather than objects with varying intrinsic importance. To address this, we propose a gravitational framework inspired by physics that interprets vector norms as mass and spatial proximity as distance. For classification, we introduce the gravitational kernel (GK)—a positive semi-definite kernel that couples norms with standard distance metrics while strictly satisfying Mercer’s condition. GK-based SVM classification is systematically evaluated through statistical hypothesis testing, computational complexity analysis, and ablation studies. For clustering, we propose input–output space vector gravitational clustering (IOSVGC), which partitions data in a standardized joint input–output embedding using deterministic force-based affinity to capture complex functional topologies—moving beyond purely geometric approaches. Beyond conventional clustering, IOSVGC is repurposed as a response-adaptive sampling strategy for surrogate modeling. Extensive experiments demonstrate consistent improvements in high-dimensional clustering and predictive fidelity for pool-based sampling tasks, proving effective across diverse data regimes.