用奇异核改进粒子采样，证明了长期极限下收敛到目标分布。
Riesz-Kernel Stein Variational Gradient Descent: Renormalized Entropy and Long-Time Particle Limits
- 移除奇异核的自相互作用，设计周期性Riesz核的SVGD方法
- 在粒子数和时间趋于无穷时，经验测度弱收敛至目标分布点质量
- 适用于需要高精度采样的机器学习与统计推断场景
Stein变分梯度下降（SVGD）通过确定性核化动力学将相互作用粒子推向目标分布。奇异Riesz核因其能提供量化群体收敛性而受关注，但在有限粒子层面，对应的Stein能量存在无限自相互作用。本文研究去除自相互作用的周期性Riesz SVGD，证明了多粒子、长时间采样定理：在奇异Stein能量局部可积的范围内，若初始每粒子相对熵有统一上界，则时间平均的经验测度律随粒子数及发散平均时间趋于无穷时弱收敛于目标分布的点质量δ_π。此外，对于有限相对熵的不变粒子律诱导的经验测度律，即使无统一熵界也收敛至δ_π。低于对数奇点阈值时，获得显式的代数有限粒子误差界。这些结果将光滑核SVGD的联合熵方法推广至奇异相互作用。
原文摘要 · Abstract (English)
Stein variational gradient descent (SVGD) transports interacting particles toward a target distribution through deterministic kernelized dynamics. Singular Riesz kernels are attractive because they can provide quantitative population-level convergence, but at the finite-particle level the corresponding Stein energy has infinite self-interaction. We study periodic Riesz SVGD with self-interaction removed and prove a many-particle, long-time sampling theorem. Throughout the range in which the singular Stein energy is locally integrable, under a uniform bound on the initial relative entropy per particle, the time-averaged empirical-measure law converges weakly to the point mass \(δ_π\) at the target as the particle number and any diverging averaging horizon tend to infinity. We also show that the empirical-measure laws induced by invariant particle laws of finite relative entropy converge weakly to \(δ_π\), without a uniform entropy bound. Below the logarithmic singularity threshold, we obtain an explicit algebraic finite-particle error bound. These results extend the joint-entropy approach for smooth-kernel SVGD to singular interactions.
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