A Note on the Convergence of Mirrored Stein Variational Gradient Descent under $(L_0,L_1)-$Smoothness Condition

In this note, we establish a descent lemma for the population limit Mirrored Stein Variational Gradient Method~(MSVGD). This descent lemma does not rely on the path information of MSVGD but rather on a simple assumption for the mirrored distribution $\mathrm{\nabla }{\mathrm{\Psi }}_{\mathrm{\#}}\pi \propto \exp (-V)$. Our analysis demonstrates that MSVGD can be applied to a broader class of constrained sampling problems with non-smooth $V$. We also investigate the complexity of the population limit MSVGD in terms of dimension $d$.