Descent modulus and applications

TL;DR

This paper introduces an axiomatic framework for descent moduli applicable to various functions on general spaces, revealing that such moduli can often determine the functions themselves, with specific insights for finite spaces.

Contribution

It proposes a unified axiomatic definition of descent modulus encompassing smooth, nonsmooth, and probabilistic functions, and demonstrates their determining power.

Findings

Descent modulus can fully determine functions in many cases.

A classification of descent operators on finite spaces is provided.

The framework unifies various notions of descent in a general setting.

Abstract

The norm of the gradient $\nabla$f (x) measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0, $\partial$f (x)) of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope |$\nabla$f |(x). In this work we propose an axiomatic definition of descent modulus T [f ](x) of a real-valued function f at every point x, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.