标题： 概率分布的几何高斯近似 显示英文标题

标题： Geometric Gaussian Approximations of Probability Distributions

摘要： 近似复杂的概率分布，如贝叶斯后验分布，在许多应用中具有中心意义。 我们研究了几何高斯近似的表达能力。 这些包括通过微分同胚或黎曼指数映射的高斯前向传播的近似。 我们首先回顾这两种不同的几何高斯近似。 然后我们探讨它们之间的关系。 我们进一步提供一个构造性证明，表明这种几何高斯近似是通用的，即它们可以捕捉任何概率分布。 最后，我们讨论给定一个概率分布族时，是否可以找到一个共同的微分同胚，以获得该族的统一高质量几何高斯近似。 显示英文摘要

摘要： 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.