Deciding when a given metric is Hessian via reparameterization and potential function

Develop a method to determine, given a Riemannian metric g expressed in a parameterization λ on a single global chart, whether there exists a reparameterization θ and a convex potential function F(θ) such that g(θ)=∇^2F(θ), thereby deciding whether g is a Hessian metric.

Background

Hessian metrics play a central role in information geometry and enable dual flat connections with powerful geometric and computational implications. The paper reviews conditions and tests for Hessian structure but highlights the lack of a general constructive method to find a suitable reparameterization and potential function when starting from a metric provided in arbitrary coordinates. Solving this would systematize identification of Hessian structures across statistical models.