Triangular maps are a construct in probability theory that allows the transformation of any source density to any target density. We consider flow based models that learn these triangular transformations which we call triangular flows and study the properties of these triangular flows with the goal of capturing heavy tailed target distributions.In one dimension, we prove that the density quantile functions of the source and target density can characterize properties of the increasing push-forward transformation and show that no Lipschitz continuous increasing map can transform a light-tailed source to a heavy-tailed target density. We further precisely relate the asymptotic behavior of these density quantile functions with the existence of certain function moments of distributions. These results allow us to give a precise asymptotic rate at which an increasing transformation must grow to capture the tail properties of a target given the source distribution. In the multivariate case, we show that any increasing triangular map transforming a light-tailed source density to a heavy-tailed target density must have all eigenvalues of the Jacobian to be unbounded. Our analysis suggests the importance of source distribution in capturing heavy-tailed distributions and we discuss the implications for flow based models.

通过研究三角形图在可计算的源密度上的作用，我们发现源密度和目标密度的分位函数是捕捉目标密度尾部所需转换的斜率的精确表征；我们进一步提出了可以同时学习源分布和三角形图来捕捉目标密度尾部属性的尾部适应流模型，以解决Real-NVP、MAF和Glow等流模型无法捕捉具有非高斯尾部的分布的问题。

利普希茨三角形流的尾部