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：以未知归一化常数采样概率分布是计算科学与工程中的基本问题。该任务可被视为在所有概率测度空间上的优化问题，通过梯度流可动态地将初始分布演化至目标极小值。以概率测度空间梯度流支配其律的平均场模型亦可被识别；这些平均场模型的粒子近似构成了算法的基础。梯度流方法也是变分推断算法的基础，其中优化在参数化概率分布族（如高斯分布）上进行，且底层梯度流被限制在该参数化族内。通过选择不同的能量泛函与梯度流度量，可得到具有不同收敛特性的算法。本文聚焦于Kullback-Leibler散度，并证明在缩放意义下，该能量选择产生的梯度流具有不依赖归一化常数的独特性质。对于度量，我们重点关注Fisher-Rao、Wasserstein与Stein度量的变体；引入梯度流及其对应平均场模型的仿射不变性性质，判定给定度量是否具有仿射不变性，若不具备则通过修正使其满足仿射不变性。我们分别在概率密度空间和高斯空间中研究由此产生的梯度流。高斯空间中的流动可视为原流动的高斯近似。我们证明了基于度量的高斯近似与矩闭合方法得到的高斯近似具有一致性，建立了二者之间的关联，并通过研究其长期收敛特性展现了仿射不变性的优势。