A priori bound for the second derivative of the optimal star-separable map with respect to the root variable

Derive an a priori bound for the second derivative with respect to the root variable z1 of the optimal star-separable transport component T⋆i(xi | z1), i≥2, under Assumptions (R), (P1), (P2), and (RD+), so that Assumption (GR) controlling the growth rate of ∂2z1 T⋆i(xi | z1) can be removed.

Background

To obtain computational guarantees, the authors introduce Assumption (GR), which posits a growth bound for the second derivative of the optimal star-separable transport map component T⋆i with respect to the root variable. While they prove a priori bounds for other derivatives, they do not establish this second-derivative bound.

Removing Assumption (GR) by proving a direct a priori bound for ∂2z1 T⋆i(xi | z1) under the paper’s regularity and domination conditions would strengthen the regularity theory and lead to sharper, assumption-free computational guarantees.

References

Whereas we have derived a priori bounds for the other derivatives of T_i\star in \cref{thm:star_caff}, the second derivative w.r.t. z is particularly difficult to control and hence we must adopt assum:GR as an assumption; we leave it for future work to obtain an a priori bound for this quantity as well.

— Theory and computation for structured variational inference  (2511.09897 - Sheng et al., 13 Nov 2025) in Section 3.4 (A projected gradient descent algorithm for solving (T-SSVI)), discussion of Assumption (GR)