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：采样具有未知归一化常数的概率分布是计算科学与工程中的基本问题。该任务可转化为概率测度空间上的优化问题，通过梯度流动动态地将初始分布演化为目标最小值。具有由概率测度空间梯度流支配律的平均场模型也可被识别；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，该方法在参数化概率分布族（如高斯分布）上执行优化，并将底层梯度流限制于该参数化族。通过选择不同的能量泛函和梯度流度量，可产生具有不同收敛特性的算法。本文聚焦于Kullback-Leibler散度，经证明在尺度变换意义下，该散度具有独特性质：由此能量选择产生的梯度流不依赖于归一化常数。在度量方面，我们重点研究Fisher-Rao度量、Wasserstein度量与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，判定给定度量能否导致仿射不变性，对不具备该性质的度量进行修改以使其实现仿射不变性。我们分别研究概率密度空间与高斯空间中的梯度流演化。高斯空间中的流可理解为流的近似形式。我们证明基于度量的高斯近似与通过矩封闭获得的高斯近似具有一致性，建立两者之间的联系，并研究其长期收敛特性，从而揭示仿射不变性的优势。