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

Let ρ0∈PV\rho_0\in\mathcal{P}_V, let SrS_r be the stated class of flows, and for u∈Sru\in S_r define the pushforward measure ρt=u(t,⋅)#ρ0\rho_t=u(t,\cdot)_{\#}\rho_0. For u u-regularized Stein Fisher information Iν,Stein⁡(ρt∣π)I_{\nu,\operatorname{Stein}}(\rho_t\mid\pi), Uniform short-time Fisher-information conjecture. For every ϵ>0\epsilon>0, there \exists T>0T>0 such that for every u∈Sru\in S_r and t∈[0,T]t\in[0,T], one has

∫0TIν,Stein⁡(ρt∣π)12 dt<ϵ.\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 ρ0∈PV\rho_0\in\mathcal{P}_V and the stated class SrS_r.

Sources & referencesView supporting material

Primary source

Ye He, Krishnakumar Balasubramanian, Bharath K. Sriperumbudur and Jianfeng Lu, “Regularized Stein Variational Gradient Flow”, arXiv:2211.07861 (2024).

fisher-informationgradient-flowsstein-variational-gradient-flow

Progress summary

Refreshed
Solved

The conjecture is stated and proved in the source paper, and no later challenge to that proof was found.

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 ρ0∈PV\rho_0\in\mathcal{P}_V, it applies uniformly to all flows u∈Sru\in S_r.

Established estimate

The paper gives T0>0T_0>0 such that, for every u∈Sru\in S_r,

∫0T0Iν,Stein⁡(ρu,t∣π)1/2 dt<ν1/2∥k∥∞−1r.\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.

Sources

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