  Given two jointly distributed random variables $(X,Y)$, a functional
representation of $X$ is a random variable $Z$ independent of $Y$, and a
deterministic function $g(\cdot, \cdot)$ such that $X=g(Y,Z)$. The problem of
finding a minimum entropy functional representation is known to be equivalent
to the problem of finding a minimum entropy coupling where, given a collection
of probability distributions $P_1, \dots, P_m$, the goal is to find a coupling
$X_1, \dots, X_m$ ($X_i \sim P_i)$ with the smallest entropy $H_\alpha(X_1,
\dots, X_m)$. This paper presents a new information spectrum converse, and
applies it to obtain direct lower bounds on minimum entropy in both problems.
The new results improve on all known lower bounds, including previous lower
bounds based on the concept of majorization. In particular, the presented
proofs leverage both - the information spectrum and the majorization -
perspectives on minimum entropy couplings and functional representations.
