Wasserstein Generative Adversarial Networks (WGANs) are the popular generative models built on the theory of Optimal Transport (OT) and the Kantorovich duality. Despite the success of WGANs, it is still unclear how well the underlying OT dual solvers approximate the OT cost (Wasserstein-1 distance, $\mathbb{W}_{1}$) and the OT gradient needed to update the generator. In this paper, we address these questions. We construct 1-Lipschitz functions and use them to build ray monotone transport plans. This strategy yields pairs of continuous benchmark distributions with the analytically known OT plan, OT cost and OT gradient in high-dimensional spaces such as spaces of images. We thoroughly evaluate popular WGAN dual form solvers (gradient penalty, spectral normalization, entropic regularization, etc.) using these benchmark pairs. Even though these solvers perform well in WGANs, none of them faithfully compute $\mathbb{W}_{1}$ in high dimensions. Nevertheless, many provide a meaningful approximation of the OT gradient. These observations suggest that these solvers should not be treated as good estimators of $\mathbb{W}_{1}$, but to some extent they indeed can be used in variational problems requiring the minimization of $\mathbb{W}_{1}$.
翻译:瓦塞斯坦 瓦塞斯坦 瓦塞斯坦 瓦塞泰因 德方网络 (WGANs) 是建立在最佳运输理论(OT) 和 Kantorovich 双重理论上流行的基因模型。 尽管WGANs取得了成功, 基础的OT 双重解决方案对于OT成本( Wasserstein-1 距离, $\ mathbb{W ⁇ 1} 美元) 和 OT 更新生成器所需的 OT 价格( OT 距离, $\ mathbb{W ⁇ 1} 美元) 仍然不清楚。 在本文中, 我们解决这些问题。 我们建了1-Lipschitz 函数, 并用它们来构建光线性单调单调运输计划。 这个战略在高空间(如图像空间等) 产生连续的基准分布配对与分析性已知的 OT 计划、 OT 成本和 OT 梯度的配对。 我们用这些基准配对进行彻底评估。 即使这些解决方案在WGANs 中表现良好,, 也没有忠实地将 $1 水平 的观察结果转化为 。