Geometry, Preference, and Shaping: A Unified View of Precision–Recall Objectives

Region-overlap objectives are usually presented as separate formulas, obscuring which differences change their optimization behavior. We organize a broad class of precision–recall scores through one construction: a weighted quasi-arithmetic mean followed by an outer transformation. Its generator, weight, and outer map specify normalized geometry, preference, and shaping. Dice, IoU, $F_{\beta }$, and Tversky share a harmonic core; arithmetic and geometric aggregation belong to the same power-mean family. The score's derivative ratio determines how it mixes precision and recall gradients, and these coefficients together with the model's metric-gradient Gram matrix determine angular discrepancies. This separates shared geometry from directional equivalence and explains why shaping can preserve each item's direction yet rotate an aggregate update. We characterize compatible ratio fields and recover the generator, preference, and outer shaping of representable scores. Controlled numerical diagnostics validate the identities and constructions. The resulting framework connects objective classification, parameter-gradient interpretation, and inverse design without equating a common representation with identical training dynamics.