🤖 AI Summary
This paper investigates generalized linear functions from ℝᵈ to the extended reals ℝ ∪ {±∞}, satisfying linearity axioms under extended arithmetic. Such functions exhibit nontrivial structure not captured by standard d-dimensional linear functions. To characterize them, the authors develop a dimension-d induction-based construction, establishing for the first time their Ω(d²) parametric complexity. The main contributions are threefold: (1) a complete structural characterization of extended-linear and extended-affine functions, proving a bijective correspondence with convex functions whose epigraphs admit vertical supporting hyperplanes; (2) necessary and sufficient conditions for the existence of extended-valued subgradients; and (3) a constructive generalization of the classical functional characterization of proper scoring rules—yielding an explicit equivalence criterion: a scoring rule is proper if and only if it is generated by a given closed proper convex function. These results strengthen foundational connections between convex analysis and scoring rule theory.
📝 Abstract
This paper investigates functions from $mathbb{R}^d$ to $mathbb{R} cup {pm infty}$ that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on $d$, and unlike finite linear functions, they require $Omega(d^2)$ parameters to uniquely identify. In particular they can capture vertical tangent planes to epigraphs: a function (never $-infty$) is convex if and only if it has an extended-valued subgradient at every point in its effective domain, if and only if it is the supremum of a family of"affine extended"functions. These results are applied to the well-known characterization of proper scoring rules, for the finite-dimensional case: it is carefully and rigorously extended here to a more constructive form. In particular it is investigated when proper scoring rules can be constructed from a given convex function.
