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This paper investigates periodic Riesz-Kernel Stein Variational Gradient Descent (SVGD) by addressing the issue of infinite self-interaction in singular Riesz kernels, which traditionally hinder finite-particle convergence. The authors establish a long-time sampling theorem that demonstrates weak convergence of the time-averaged empirical measure to a point mass at the target distribution, under specific conditions on initial relative entropy. Additionally, they provide an explicit algebraic error bound for finite-particle scenarios below the logarithmic singularity threshold, thereby extending the applicability of SVGD to singular interactions.
Weak convergence to target distributions can be achieved even with singular Riesz kernels, challenging previous assumptions about self-interaction in particle dynamics.
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 \(\delta_\pi\) 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 \(\delta_\pi\), 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.
