Scaling-invariant functions versus positively homogeneous functions

TL;DR

This paper characterizes scaling-invariant functions as compositions of monotonic functions with positively homogeneous functions, providing necessary and sufficient conditions and exploring their sublevel sets.

Contribution

It establishes a comprehensive characterization of scaling-invariant functions, linking them to positively homogeneous functions and monotonic transformations, with generalizations of sublevel set properties.

Findings

Necessary and sufficient conditions for scaling-invariant functions to be composites.

Generalization of sublevel set properties of positively homogeneous functions.

Insight into the structure of scaling-invariant functions.

Abstract

Scaling-invariant functions preserve the order of points when the points are scaled by the same positive scalar (with respect to a unique reference point). Composites of strictly monotonic functions with positively homogeneous functions are scaling-invariant with respect to zero. We prove in this paper that the reverse is true for large classes of scaling-invariant functions. Specifically, we give necessary and sufficient conditions for scaling-invariant functions to be composites of a strictly monotonic function with a positively homogeneous function. We also study sublevel sets of scaling-invariant functions generalizing well-known properties of positively homogeneous functions.