目标自适应Green-Bessel SVGD：混沌传播的时间一致性与最后迭代一致性
Target-adapted Green-Bessel SVGD: uniform-in-time propagation of chaos and last-iterate consistency
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- Rice University（莱斯大学）
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中文总结 AI 辅助
本文针对紧致流形上的目标自适应SVGD流，证明其混沌传播和最后迭代具有时间一致性，期望均匀差异为O((log N)^(-1/2))，并给出有限模态逼近定理。
中文摘要 AI 辅助
我们证明了在紧致连通流形上，目标自适应Stein变分梯度下降（SVGD）流具有时间一致性的混沌传播和最后迭代一致性。目标具有光滑正密度，粒子从固定的光滑非负密度比独立初始出发。该构造使用可逆目标Langevin生成元的Green-Bessel算子$Q_{r,\pi}=A_\pi^{-1}(\mathrm{Id}+A_\pi)^{-r}$。充分的Bessel平滑给出具有有限对角线的标量核，正矩阵提升将其势力实现为Stein速度。群体和经验流随后耗散相同的有限目标差异。群体熵和目标谱隙给出该差异的衰减；有限时间粒子比较达到一个时间点，此后共同能量单调性控制所有后续时间。所得期望均匀差异为$O((\log N)^{-1/2})$，具有相应的对数$W_1$界，并沿每个序列$t_N\to\infty$保持一致。我们还证明了精确的有限模态逼近定理，具有显式空间分辨率误差和特征因子化粒子实现。对于约束欧几里得目标，我们建立了静态核和矩结果，并在显式群体正则性和输运假设下给出条件动态扩展。
英文摘要
We prove uniform-in-time propagation of chaos and last-iterate consistency for a target-adapted Stein variational gradient descent (SVGD) flow on compact connected manifolds. The target has a smooth positive density, and the particles start independently from a fixed smooth nonnegative density ratio. The construction uses the Green--Bessel operator $Q_{r,π}=A_π^{-1}(\mathrm{Id}+A_π)^{-r}$ of the reversible target Langevin generator. Sufficient Bessel smoothing gives a scalar kernel with finite diagonal, and a positive matrix lift realizes its potential force as a Stein velocity. Population and empirical flows then dissipate the same finite target discrepancy. Population entropy and the target spectral gap give decay of this discrepancy; a finite-time particle comparison reaches a time after which common-energy monotonicity controls every later time. The resulting expected uniform discrepancy is $O((\log N)^{-1/2})$, with a corresponding logarithmic $W_1$ bound and consistency along every sequence $t_N\to\infty$. We also prove an exact finite-mode approximation theorem with an explicit spatial-resolution error and a feature-factorized particle implementation. For confining Euclidean targets, we establish static kernel and moment results and give a conditional dynamical extension under explicit population-regularity and transport hypotheses.
