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

Maximum likelihood (ML) learning for energy-based models (EBMs) is
challenging, partly due to non-convergence of Markov chain Monte Carlo.Several
variations of ML learning have been proposed, but existing methods all fail to
achieve both post-training image generation and proper density estimation. We
propose to introduce diffusion data and learn a joint EBM, called diffusion
assisted-EBMs, through persistent training (i.e., using persistent contrastive
divergence) with an enhanced sampling algorithm to properly sample from
complex, multimodal distributions. We present results from a 2D illustrative
experiment and image experiments and demonstrate that, for the first time for
image data, persistently trained EBMs can {\it simultaneously} achieve long-run
stability, post-training image generation, and superior out-of-distribution
detection.

本文介绍了通过引入扩散数据和使用增强采样算法，通过持续对比散度学习一种联合的能量基模型，称为扩散辅助 - EBM，来实现复杂的，多模态分布的准确采样和密度估计，并进行了 2D 和图像实验，证明了持续训练的 EBMs 可以同时实现长时间稳定性，后训练图像生成和卓越的异常检测。