Gradient flows, second order gradient systems and convexity

TL;DR

This paper explores the relationship between gradient flows and second order gradient systems for convex functions, revealing that equal gradient norms imply the functions differ only by a constant.

Contribution

It establishes a novel connection between gradient flows and second order systems, leading to a uniqueness result for convex functions with equal gradient norms.

Findings

Gradient flow of a smooth function relates to second order gradient system orbits.

Equal gradient norms for convex functions imply the functions differ by a constant.

Provides new insights into the structure of convex functions and their gradient behaviors.

Abstract

We disclose an interesting connection between the gradient flow of a $\mathcal{C}^2$-smooth function $\psi$ and evanescent orbits of the second order gradient system defined by the square-norm of $\nabla\psi$, under adequate convexity assumption. As a consequence, we obtain the following surprising result for two $\mathcal{C}^2$, convex and bounded from below functions $\psi_1$, $\psi_2$: if $||\nabla\psi_1||=||\nabla\psi_2||$, then $\psi_1=\psi_2 + k$, for some $k\in \mathbb{R}$.