Convex functions with symplectic Hessian

What the math gives to ML

The paper gives an intrinsic interior curvature estimate for Hessian metrics whose Hessian is both positive definite and symplectic: the scalar curvature is nonnegative and decays as the inverse square of the metric distance to the boundary. This is transferable to input-convex neural potentials as a geometric regularizer: enforce the algebraic constraint H J H = J on the network Hessian and penalize the resulting Hessian-metric curvature. The theorem supplies a falsifiable stability certificate rather than merely a heuristic smoothness penalty: exact symplectic convex potentials on all of R^{2m} must be quadratic, so nonquadratic learned behavior should be localized near the effective domain boundary. The main limitation is computational cost, since curvature requires third derivatives and the theorem assumes exact rather than approximate symplecticity.