用几何变换的高斯分布逼近任意复杂概率分布，理论证明其通用性。
Geometric Gaussian Approximations of Probability Distributions
- 通过微分同胚或黎曼指数映射将高斯分布变形以逼近目标分布
- 证明此类几何高斯逼近具有普适性，可表示任意概率分布
- 适用于贝叶斯后验等复杂分布建模，适合理论与推断研究者
近似复杂概率分布（如贝叶斯后验）在众多应用中至关重要。本文研究几何高斯逼近的表达能力，这类逼近通过微分同胚或黎曼指数映射对高斯分布进行变形。首先回顾这两种几何高斯逼近方法，探讨其相互关系；进一步提供构造性证明，表明此类逼近具有普遍性，可精确捕捉任意概率分布。最后讨论：对于给定的概率分布族，是否存在一个通用的微分同胚，使得该族中所有分布都能获得高质量的几何高斯逼近。
原文摘要 · Abstract (English)
Approximating complex probability distributions, such as Bayesian posterior distributions, is of central interest in many applications. We study the expressivity of geometric Gaussian approximations. These consist of approximations by Gaussian pushforwards through diffeomorphisms or Riemannian exponential maps. We first review these two different kinds of geometric Gaussian approximations. Then we explore their relationship to one another. We further provide a constructive proof that such geometric Gaussian approximations are universal, in that they can capture any probability distribution. Finally, we discuss whether, given a family of probability distributions, a common diffeomorphism can be found to obtain uniformly high-quality geometric Gaussian approximations for that family.
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