发表机构
Rice University（里士大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究去除自相互作用的周期里斯SVGD，证明多粒子长时间采样定理。在奇异斯坦能量局部可积且初始粒子相对熵有界时，时间平均经验测度律弱收敛到目标点质量。还表明有限相对熵不变粒子律诱导的经验测度律也收敛，且在对数奇点阈值下有有限粒子误差界。
AI 中文摘要
斯坦变分梯度下降（SVGD）通过确定性核动力将相互作用粒子输运至目标分布。奇异里斯核很有吸引力，因其能提供定量总体收敛，但有限粒子层面相应斯坦能量有无限自相互作用。我们研究去除自相互作用的周期里斯SVGD并证明多粒子长时间采样定理。在奇异斯坦能量局部可积范围内，当初始粒子相对熵有一致界时，时间平均经验测度律在粒子数和平均视界趋于无穷时弱收敛到目标处的点质量\(\delta_\pi\)。我们还表明有限相对熵不变粒子律诱导的经验测度律弱收敛到\(\delta_\pi\)，无需一致熵界。在对数奇点阈值以下，我们得到显式代数有限粒子误差界。这些结果将光滑核SVGD的联合熵方法扩展到奇异相互作用。
英文摘要
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.
Comments31 pages. No figures. Many-particle and long-time convergence for Stein variational gradient descent with singular periodic Riesz kernels
