标题： 卡尔曼-朗之万动力学：指数收敛、粒子近似和数值近似 显示英文标题

标题： Kalman-Langevin dynamics : exponential convergence, particle approximation and numerical approximation

摘要： 兰姆方程动力学在采样、优化和估计方面找到了大量应用。用协方差预处理动力学中的梯度——这一思想起源于利用卡尔曼技术解决估计和反问题的相关文献——导致了一个平均场（McKean-Vlasov）随机微分方程的产生。 我们证明了平均场随机微分方程的时间边缘分布律在非高斯势能下收敛到吉布斯测度，并且这种收敛是指数级的。这将之前在高斯环境下得到的结果扩展到了更广泛的势函数类别。 此外，我们还建立了所有矩在时间上的均匀界，并证明了在 $p$-Wasserstein 距离下的收敛性。进一步地，我们展示了弱粒子近似方法的收敛性，该方法避免了计算经验协方差矩阵的平方根，从而收敛到平均场极限。最后，我们证明了一种用于逼近粒子动力学的显式数值方案，在粒子数量趋于无穷时，收敛到其连续时间极限，解决了测度中的非全局Lipschitz性问题。 显示英文摘要

摘要： Langevin dynamics has found a large number of applications in sampling, optimization and estimation. Preconditioning the gradient in the dynamics with the covariance - an idea that originated in literature related to solving estimation and inverse problems using Kalman techniques - results in a mean-field (McKean-Vlasov) SDE. We demonstrate exponential convergence of the time marginal law of the mean-field SDE to the Gibbs measure with non-Gaussian potentials. This extends previous results, obtained in the Gaussian setting, to a broader class of potential functions. We also establish uniform in time bounds on all moments and convergence in $p$-Wasserstein distance. Furthermore, we show convergence of a weak particle approximation, that avoids computing the square root of the empirical covariance matrix, to the mean-field limit. Finally, we prove that an explicit numerical scheme for approximating the particle dynamics converges, uniformly in number of particles, to its continuous-time limit, addressing non-global Lipschitzness in the measure.