Threshold Metric Spaces for Deep Metric Learning

Deep metric learning thresholds a learned distance to decide when two inputs match, and the resulting relation is nontransitive: A may match B and B match C , whereas A and C are rejected. No classical metric‐type space can represent this, since in each the zero‐distance relation is forced to be transitive. We introduce the φ ‐threshold metric space, which relaxes the triangle inequality by an additive correction φ with φ (0, 0) = k > 0, and prove that a metric‐type space admits a nontransitive zero‐distance relation if and only if φ (0, 0) > 0. It is thus the first metric‐type structure modeling thresholded learned similarity. We show that every thresholded learned pseudometric is a φ ‐threshold metric through the dead‐zone construction d _τ = max{0, D − τ }, identify the contrastive and triplet margins with k , prove that the verification relation is a tolerance that no re‐embedding can make transitive, and bound transitive‐closure identity drift by ( n − 1) τ over n hops. A representation theorem gives semiorders and tolerance spaces the quantitative distance they lacked. We develop the topology, completeness, and a Banach type fixed point theorem, and apply the theory to perceptual quantization. The just‐noticeable difference paradox is the classical one‐dimensional instance.