Let rho_0∈P_V, let S_r be the stated class of flows, and for u∈ S_r define the pushforward measure rho_t=u(t,·)_#rho_0. For u-regularized Stein Fisher information I_nu,operatornameStein(rho_t|pi), Uniform short-time Fisher-information conjecture. For every epsilon>0, there exists T>0 such that for every u∈ S_r and t∈[0,T], one has int_0^T I_nu,operatornameStein(rho_t|pi)^frac12,dt<epsilon. This assertion is used to control the short-time behavior of the regularized Stein variational gradient flow and support the local-to-global existence argument. The source provides no resolution or further conditions beyond rho_0∈P_V and the stated class S_r.

Uniform short-time Fisher-information bound for regularized Stein variational flows

Let $\rho_0\in\mathcal{P}_V$ , let $S_r$ be the stated class of flows, and for $u\in S_r$ define the pushforward measure $\rho_t=u(t,\cdot)_{\#}\rho_0$ . For $u$ -regularized Stein Fisher information $I_{\nu,\operatorname{Stein}}(\rho_t\mid\pi)$ , Uniform short-time Fisher-information conjecture. For every $\epsilon>0$ , there \exists $T>0$ such that for every $u\in S_r$ and $t\in[0,T]$ , one has

\int_0^T I_{\nu,\operatorname{Stein}}(\rho_t\mid\pi)^{\frac{1}{2}}\,dt<\epsilon.

This assertion is used to control the short-time behavior of the regularized Stein variational gradient flow and support the local-to-global existence argument. The source provides no resolution or further conditions beyond $\rho_0\in\mathcal{P}_V$ and the stated class $S_r$ .

The assertion appears as Lemma 2 of “Regularized Stein Variational Gradient Flow” and is used in the local-existence and global weak-solution theory. Under the paper’s assumptions and $\rho_0\in\mathcal{P}_V$ , it applies uniformly to all flows $u\in S_r$ .

Established estimate

The paper gives $T_0>0$ such that, for every $u\in S_r$ ,

\int_0^{T_0} I_{\nu,\operatorname{Stein}}(\rho_{u,t}\mid\pi)^{1/2}\,dt<\nu^{1/2}\lVert k\rVert_\infty^{-1}r.

Publication and subsequent scan

The work was reported as accepted in Foundations of Computational Mathematics. The retrieved literature contains no counterexample, correction, withdrawal, independent gap report, or competing resolution concerning this bound.

Current status (as of August 2026): the bound is proved in the source paper and supports its local-to-global existence argument; no unresolved mathematical objection was found.

Uniform short-time Fisher-information bound for regularized Stein variational flows

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