  We investigate the ability of popular flow based methods to capture
tail-properties of a target density by studying the increasing triangular maps
used in these flow methods acting on a tractable source density. We show that
the density quantile functions of the source and target density provide a
precise characterization of the slope of transformation required to capture
tails in a target density. We further show that any Lipschitz-continuous
transport map acting on a source density will result in a density with similar
tail properties as the source, highlighting the trade-off between a complex
source density and a sufficiently expressive transformation to capture
desirable properties of a target density. Subsequently, we illustrate that flow
models like Real-NVP, MAF, and Glow as implemented originally lack the ability
to capture a distribution with non-Gaussian tails. We circumvent this problem
by proposing tail-adaptive flows consisting of a source distribution that can
be learned simultaneously with the triangular map to capture tail-properties of
a target density. We perform several synthetic and real-world experiments to
compliment our theoretical findings.
