Evolutionary Adaptive Metric Learning: A Multi-Objective Framework for Learning Local Distance Functions in Nearest Neighbor Classification

Isaac Touza

Nearest neighbour classifiers inherit the geometry of their distance function. Fixed distances and globally learned metrics apply the same geometry throughout the input space, although relevant feature directions may vary across regions. We propose EAML (Evolutionary Adaptive Metric Learning), a framework that learns instance-dependent positive-definite metrics through a low-dimensional parameterization. Each local metric combines an input-dependent diagonal scaling, generated by softmax interpolation between anchor points, with a global low-rank correlation component. A real-coded elitist genetic algorithm optimizes the parameters using a scalar objective combining leave-one-out prediction error, robustness, metric smoothness, and regularization. We establish positive definiteness, metric equivalence, existence of an optimal solution, and Lipschitz continuity of the metric field. We further investigate prediction stability, generalization bounds, and evolutionary convergence under explicit assumptions. Experiments on synthetic data with region-dependent feature relevance show that EAML can recover local metric variations and outperform global metrics. Evaluation on thirteen…