Continuous Sensitivity and Reversibility

TL;DR

This paper introduces the concept of continuous sensitivity, an invariant that quantifies how sensitive a differentiable function is to input changes, aiding in reversibility analysis without explicitly computing inverses.

Contribution

The paper defines continuous sensitivity as a new invariant for differentiable functions, providing a practical tool to assess reversibility and sensitivity without explicit inverse functions.

Findings

Continuous sensitivity ranges from 0 to n, indicating the degree of input sensitivity.

A value of 0 corresponds to constant functions; n indicates injective functions.

Tools for computing continuous sensitivity are developed, facilitating reversibility analysis.

Abstract

Let $n$ be a positive integer and $f$ a differentiable function from a convex subset $C$ of the Euclidean space $\mathbb{R}^n$ to a smooth manifold. We define an invariant of $f$ via counting certain threshold functions associated to $f$. We call this invariant the continuous sensitivity of $f$ and denote it by $\mathrm{cs}_{C}(f)$. This invariant is a real number between $0$ and $n$ and measures how sensitive $f$ is to change in its input variables. For example, if $f$ is a constant function then $\mathrm{cs}_{C}(f)=0$. On the other extreme, if $\mathrm{cs}_{C}(f)=n$ then $f$ is one-to-one on $C$. This last statement is important for reversibility problems. To say that a function is reversible one can write an explicit inverse of the function. However, this is not always easy. Even a multilinear function can have a complicated inverse function. Here we give tools to compute continuous sensitivity which makes it possible to answer reversibility problems without finding explicit inverse functions.