Optimization landscape and feasibility in the updated Riemannian AmbientFlow objective

Ascertain which local minimum is attained when optimizing the updated Riemannian AmbientFlow objective that minimizes the AmbientFlow variational lower bound with the added geometric regularization term penalizing the Frobenius norm of the Jacobian of the learned diffeomorphism at the origin, and determine whether the feasibility assumptions used in the recoverability theorem—namely, the existence of parameters such that the learned data distribution equals the ground-truth data distribution and the learned posterior equals the true posterior while satisfying the geometric constraint—hold for minimizers of this objective.

Background

The paper introduces Riemannian AmbientFlow, which augments the AmbientFlow variational lower bound with a geometric regularization derived from pullback Riemannian geometry to encourage low-dimensional manifold structure. A theoretical recoverability result is proved under assumptions that include the existence of parameters achieving exact data and posterior matching and satisfying a geometric constraint.

However, the authors note practical caveats: the optimization is nonconvex, and it is not guaranteed which local minimum training will reach, nor whether the feasibility assumptions required by the recoverability theorem are met at the attained solution. This raises an unresolved question about the optimization landscape and the validity of the feasibility assumptions for practical minimizers.