Linear Functions to the Extended Reals

TL;DR

This paper characterizes extended real-valued linear functions, explores their structure and parameters, and applies these insights to the theory of proper scoring rules in convex analysis.

Contribution

It provides a detailed structural analysis of extended real-valued linear functions and extends the characterization of proper scoring rules to a more constructive framework.

Findings

Extended real-valued linear functions require a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0 parameters to be uniquely identified.

They can capture vertical tangent planes to epigraphs and characterize convex functions via subgradients.

The work extends the characterization of proper scoring rules to a more constructive and rigorous framework.

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.