This paper approaches the unsupervised learning problem by minimizing the
second-order Wasserstein loss (the $W_2$ loss). The minimization is
characterized by a distribution-dependent ordinary differential equation (ODE),
whose dynamics involves the Kantorovich potential between a current estimated
distribution and the true data distribution. A main result shows that the
time-marginal law of the ODE converges exponentially to the true data
distribution. To prove that the ODE has a unique solution, we first construct
explicitly a solution to the associated nonlinear Fokker-Planck equation and
show that it coincides with the unique gradient flow for the $W_2$ loss. Based
on this, a unique solution to the ODE is built from Trevisan's superposition
principle and the exponential convergence results. An Euler scheme is proposed
for the distribution-dependent ODE and it is shown to correctly recover the
gradient flow for the $W_2$ loss in the limit. An algorithm is designed by
following the scheme and applying persistent training, which is natural in our
gradient-flow framework. In both low- and high-dimensional experiments, our
algorithm converges much faster than and outperforms Wasserstein generative
adversarial networks, by increasing the level of persistent training
appropriately.

通过最小化二阶 Wasserstein 损失（即 $W_2$ 损失），该论文处理无监督学习问题。论文证明了方式一通过分布相关的常微分方程（ODE）动力学的超限势潜力近似估计当前分布与真实数据分布之间的关系。主要结果显示 ODE 的时变边界概率收敛到真实数据分布。为了证明 ODE 具有唯一解，首先明确构造了与关联的非线性 Fokker-Planck 方程相关的解，并证明它与 $W_2$ 损失的唯一梯度流相吻合。基于此，通过 Trevisan 的叠加原理和指数收敛结果，构建了 ODE 的唯一解。该论文提出了一个分布相关 ODE 的欧拉方案，并在极限情况下正确恢复了 $W_2$ 损失的梯度流。通过遵循该方案和应用持久训练，设计了一个算法，其自然地适用于梯度流框架。在低维和高维实验中，我们的算法通过适当增加持久训练水平，比 Wasserstein 生成对抗网络收敛更快且性能更好。

通过最小化 Wasserstein-2 损失进行生成建模

Generative Modeling by Minimizing the Wasserstein-2 Loss

The latent space of GANs contains rich semantics reflecting the training
data. Different methods propose to learn edits in latent space corresponding to
semantic attributes, thus allowing to modify generated images. Most supervised
methods rely on the guidance of classifiers to produce such edits. However,
classifiers can lead to out-of-distribution regions and be fooled by
adversarial samples. We propose an alternative formulation based on the
Wasserstein loss that avoids such problems, while maintaining performance
on-par with classifier-based approaches. We demonstrate the effectiveness of
our method on two datasets (digits and faces) using StyleGAN2.

本文提出了一种基于 Wasserstein 损失和 StyleGAN2 的生成对抗网络（GANs）的方法，可以实现在潜空间中编辑语义属性以修改生成的图片，避免了定向分类器会遇到的问题，并且取得了与分类器方法相当的性能。