Ensemble Kalman Sampling: mean-field limit and convergence analysis
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Ensemble Kalman sampling (EKS) is a method to find i.i.d. samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unknown. In this paper, we analyze the continuous version of EKS, a coupled SDE system, and justify its mean-filed limit is a Fokker-Planck equation, whose equilibrium state is the target distribution. This proves that in long time, the samples generated by EKS indeed are approximately i.i.d. samples from the target distribution. We further show the ensemble distribution of EKS converges, in Wasserstein-2 sense, to the target distribution with a near-optimal rate. We emphasize that even when the forward map is linear, due to the ensemble nature of the method, the SDE system and the corresponding Fokker-Planck equation are still nonlinear.
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