Sampling a probability distribution with an unknown normalization constant is a fundamental problem in computational science and engineering. This task may be cast as an optimization problem over all probability measures, and an initial distribution can be evolved to the desired minimizer dynamically via gradient flows. Mean-field models, whose law is governed by the gradient flow in the space of probability measures, may also be identified; particle approximations of these mean-field models form the basis of algorithms. The gradient flow approach is also the basis of algorithms for variational inference, in which the optimization is performed over a parameterized family of probability distributions such as Gaussians, and the underlying gradient flow is restricted to the parameterized family. By choosing different energy functionals and metrics for the gradient flow, different algorithms with different convergence properties arise. In this paper, we concentrate on the Kullback-Leibler divergence after showing that, up to scaling, it has the unique property that the gradient flows resulting from this choice of energy do not depend on the normalization constant. For the metrics, we focus on variants of the Fisher-Rao, Wasserstein, and Stein metrics; we introduce the affine invariance property for gradient flows, and their corresponding mean-field models, determine whether a given metric leads to affine invariance, and modify it to make it affine invariant if it does not. We study the resulting gradient flows in both probability density space and Gaussian space. The flow in the Gaussian space may be understood as a Gaussian approximation of the flow. We demonstrate that the Gaussian approximation based on the metric and through moment closure coincide, establish connections between them, and study their long-time convergence properties showing the advantages of affine invariance.

翻译：采样一个归一化常数未知的概率分布是计算科学与工程中的基本问题。该任务可被表述为所有概率测度上的优化问题，初始分布通过梯度流动态演化至期望的最小值。其律由概率测度空间中的梯度流支配的均值场模型也可被识别；这些均值场模型的粒子近似构成了算法的基础。梯度流方法同样是变分推断算法的基础，其中优化是在参数化概率分布族（如高斯分布）上进行的，且底层梯度流被限制在该参数化族中。通过选择不同的能量泛函和度量用于梯度流，会产生具有不同收敛特性的算法。本文在证明KL散度（在缩放意义下）具有独特性质——即由此能量选择产生的梯度流不依赖于归一化常数——后，重点关注该散度。对于度量，我们聚焦于Fisher-Rao度量、Wasserstein度量和Stein度量的变体；我们引入梯度流及其对应均值场模型的仿射不变性，判断给定度量是否导致仿射不变性，并针对不具有该性质的度量进行修改以使其具备仿射不变性。我们研究了概率密度空间和高斯空间中的梯度流。高斯空间中的流可被理解为流的近似。我们证明了基于度量与基于矩封闭的高斯近似的一致性，建立了它们之间的联系，并通过研究其长期收敛性质展示了仿射不变性的优势。