# Linear Functions to the Extended Reals

arXiv: https://arxiv.org/abs/2102.09552 · HTML: https://mlsanity.com/paper/2102.09552/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Feb 2021.

## Built on by (most cited first, 1 of 1)

- [Comparing Sequential Forecasters](https://mlsanity.com/paper/2110.00115/) (2021)

## Abstract (arXiv)

This paper investigates functions from ℝd to ℝ cup ± ∞ 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(d2) parameters to uniquely identify. In particular they can capture vertical tangent planes to epigraphs: a function (never -∞) 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.