EHC k-NN: Elliptic hypercomplex distance metrics for dimension-adaptive k-nearest neighbor

This study introduces a dimension-adaptive k-Nearest Neighbor (k-NN) model that employs a family of elliptic hypercomplex distance metrics, addressing the limitations of Euclidean geometry in heterogeneous and correlated data with tabular and image datasets. The approach reshapes the feature space using a negative real parameter $p<0$, enabling curvature-controlled neighborhoods that better capture local structure. In the proposed method, each data instance is represented as an $n$-dimensional elliptic hypercomplex number, and distances are computed through a norm that re-weights even- and odd- indexed components depending on $p$. The proposed method is dimension-adaptive in the sense that each real-valued feature vector of length $d$ is mapped to the smallest elliptic hypercomplex algebra of dimension $2^m$ satisfying $2^{m-1}<d\le 2^m$. When $d\ne 2^m$, the remaining components are zero-padded, so distance computations are carried out consistently in the corresponding $2^m$-dimensional elliptic hypercomplex space. Experiments were conducted on five tabular UCI + two image-derived benchmarks selected for their diversity in feature types and class structure. Performance was evaluated using classification performance evaluation metrics under identical $k$ settings. The proposed metric yields clear gains over Euclidean k-NN, particularly in Wine (approximately $2.0$-$2.3\%$) and Breast Cancer (approximately $1.4\%$). Improvements are moderate in Car Evaluation, while Iris and Banknote Authentication exhibit minimal change due to saturated separability and dominant attributes. On image-derived benchmarks (Seeds/Wheat and Image Segmentation), the proposed metric also delivers consistent improvements, typically around +2.0-2.7% in accuracy and +2.4-2.9% in F1-score compared with Euclidean k-NN. Further comparisons against metric-learning and manifold-inspired baselines (LMNN and geodesic distance) indicate that the proposed hypercomplex metric remains competitive and stable across neighborhood sizes, reinforcing its robustness beyond Euclidean geometry. Overall, the results indicate that the performance gains stem from the $p$-induced anisotropy of the elliptic hypercomplex norm, which reshapes neighborhood geometry to better align with heterogeneous and correlated feature structures, thus positioning elliptic hypercomplex k-NN as a robust alternative to Euclidean k-NN.