发表机构
Department of Mathematics, UCLA（加州大学洛杉矶分校数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对仿射不变集合朗之万动力学及其离散变体，通过构造新型李雅普诺夫函数解决其几何遍历性的开放问题，证明了正则化后方法的收敛性及离散化不变分布的弱收敛性。
AI 中文摘要
仿射不变集合采样器广泛应用于贝叶斯问题中，但它们的定量收敛理论，尤其是几何遍历性，仍是一个基础的开放问题。我们研究仿射不变集合朗之万动力学，这是一种将整个集合的经验协方差作为预条件子的相互作用粒子系统。尽管该方法在实践中有效，但除了总变差下的定性收敛外，目前缺乏对其的理论理解；一个核心难点在于经验协方差可能趋近于奇异。本文解决了这一挑战：对于具有有界Hessian且在球外强凸的势函数，我们结合逆协方差障碍与强制指数能量，利用新型李雅普诺夫函数证明了几何遍历性。随后，我们表明直接应用欧拉-丸山（Euler--Maruyama）格式可能以正概率发散，即使是针对一维高斯目标的情况。这促使我们提出协方差迹时间正则化，我们证明了正则化扩散的几何遍历性，且当步长足够小时，其未校正欧拉-丸山离散化也具有几何遍历性。我们还表明，当步长趋于零时，离散化的不变分布弱收敛到乘积目标分布。
英文摘要
Affine-invariant ensemble samplers are widely used in Bayesian applications. However, their quantitative convergence theory, in particular geometric ergodicity, remains a basic open question. We study the affine invariant ensemble Langevin dynamics, an interacting particle system that uses the empirical covariance of the whole ensemble as a preconditioner. While effective in practice, theoretical understanding of this method is not available beyond plain qualitative convergence in total variation, with a central difficulty being that the empirical covariance can become degenerate. For potentials with bounded Hessian that are strongly convex outside a ball, we prove geometric ergodicity using a novel Lyapunov function that combines the inverse covariance with a coercive exponential energy. We then show that directly applying the Euler--Maruyama scheme can diverge with positive probability, even for a one-dimensional Gaussian target. This motivates a covariance-trace regularization in continuous time. We prove geometric ergodicity of the regularized diffusion and, for sufficiently small step size, of its unadjusted Euler--Maruyama discretization. We also show that the invariant distributions of the discretization converge weakly to the product target distribution as the step size tends to zero. Lastly, we provide an explicit discretization that is both affine-invariant and geometrically ergodic, based on an adaptive step size clamping.
