Continuous Sensitivity and Reversibility

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.