Principled selection of the convex potential function

Develop a principled theoretical procedure for selecting the optimal convex potential function \(\phi\) in generalized score matching, including its boundary-attenuation behavior, to improve finite-sample estimator performance.

Background

The generalized score matching framework defines the generator $G_{\phi}$ through the Hessian of a strictly convex function $\phi$. Different choices of $\phi$, including power, entropic, and logarithmic barriers, induce different rates of attenuation near the boundary of a convex support and produce substantially different empirical estimation errors.

Although the experiments demonstrate that some choices perform better than others, they do not provide a theoretical rule for selecting $\phi$ before observing parameter-estimation performance. The unresolved problem is therefore to determine an principled selection procedure for the potential function, motivated by the observed dependence of finite-sample behavior on boundary attenuation.