An invertible function is bi-Lipschitz if both the function and its inverse have bounded Lipschitz constants. Nowadays, most Normalizing Flows are bi-Lipschitz by design or by training to limit numerical errors (among other things). In this paper, we discuss the expressivity of bi-Lipschitz Normalizing Flows and identify several target distributions that are difficult to approximate using such models. Then, we characterize the expressivity of bi-Lipschitz Normalizing Flows by giving several lower bounds on the Total Variation distance between these particularly unfavorable distributions and their best possible approximation. Finally, we discuss potential remedies which include using more complex latent distributions.

翻译：如果函数及其反向都连接了利普西茨常数,那么一个不可忽略的功能就是双利普西茨。 如今,多数正常化流动都是双利普西茨,通过设计或培训限制数字错误(等等),我们在本文件中讨论双利普西茨的标准化流动的表达性,并确定难以使用这种模型的几种目标分布。然后,我们通过对这些特别不受欢迎的分布与其最佳近似值之间的总变化距离给予几个较低的界限来描述双利普西茨的正常化流动的表达性。 最后,我们讨论潜在的补救措施,包括使用更复杂的潜在分布。