Differentiability of the SSVI minimizer

Establish differentiability of the unique star-structured variational inference minimizer π⋆ (i.e., show that π⋆ admits a differentiable density) under the paper’s standing assumptions on the target posterior π, including C2 regularity and strong log-concavity. This is needed to justify the first-order optimality and self-consistency equations derived for π⋆ without assuming differentiability a priori.

Background

The self-consistency equations are obtained via first-order optimality in the Wasserstein geometry, which requires the minimizer to be differentiable. The authors therefore assume differentiability of π⋆ to proceed, but do not prove it under their log-concavity framework.

A proof that the KL projection onto the star variational family C_star yields a minimizer with a differentiable density (under the stated assumptions on π) would remove this technical assumption and strengthen the theoretical foundation of their fixed-point characterization and ensuing regularity results.

References

From this point onward, we will implicitly assume that \pi\star is differentiable in all subsequent results. Since our focus here is to characterize the structure of the minimizer, we leave verification of this assumption to future work.

— Theory and computation for structured variational inference  (2511.09897 - Sheng et al., 13 Nov 2025) in Section 2.2 (Regularity and approximation quality via self-consistency equations), after Theorem self-consistency