The paper analyzes the convergence control properties of kernel Stein discrepancies (KSDs). It first shows that standard KSDs used for weak convergence control fail to control moment convergence. To address this limitation, the authors provide sufficient conditions under which alternative diffusion KSDs control both moment and weak convergence.
The key insights are:
Standard KSDs with kernels previously recommended for weak convergence control cannot enforce q-Wasserstein convergence, which is a stronger mode of convergence that controls expectations of polynomially growing continuous functions.
The authors establish sufficient conditions on the kernel to guarantee that the KSD controls both q-Wasserstein convergence and weak convergence. Specifically, they show that using a kernel of a particular form, the KSD is equivalent to q-Wasserstein convergence.
Under additional assumptions, the authors obtain an explicit upper bound on the q-Wasserstein distance in terms of the KSD, demonstrating the rate of q-Wasserstein convergence relative to KSD convergence.
The results provide a theoretical foundation for using the KSD as a quality measure for distribution approximation, especially when dealing with unbounded functions of polynomial growth.
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by Heishiro Kan... at arxiv.org 10-01-2024
https://arxiv.org/pdf/2211.05408.pdfDeeper Inquiries