A Geometric Variational View of Competitive Boundaries in Metric Learning

We study metric learning in tasks where predictions are selected from a learned score landscape and correctness is defined by task-specific tolerances. We develop a geometric bridge from these tolerances to parameter-space learning directions, providing a task-derived reference for evaluating comparison-based updates. Specifically, we quantify the extent of acceptable hypotheses that outrank all unacceptable alternatives, turning the model’s realization of task tolerance into a regional objective. Applying Hadamard’s first-variation formula, we derive the steepest direction for first-order regional expansion and connect it to the direction governing region formation. These directions allow the task utility of individual training comparisons and their aggregation by metric losses to be assessed within a common framework. Empirical analyses on real-world datasets support the theoretical predictions and link regional expansion to improved task performance. Together, these results establish a geometric bridge between metric-learning updates and task-defined regions of correct prediction.