  We point out a limitation of the mutual information neural estimation (MINE)
where the network fails to learn at the initial training phase, leading to slow
convergence in the number of training iterations. To solve this problem, we
propose a faster method called the mutual information neural entropic
estimation (MI-NEE). Our solution first generalizes MINE to estimate the
entropy using a custom reference distribution. The entropy estimate can then be
used to estimate the mutual information. We argue that the seemingly redundant
intermediate step of entropy estimation allows one to improve the convergence
by an appropriate reference distribution. In particular, we show that MI-NEE
reduces to MINE in the special case when the reference distribution is the
product of marginal distributions, but faster convergence is possible by
choosing the uniform distribution as the reference distribution instead.
Compared to the product of marginals, the uniform distribution introduces more
samples in low-density regions and fewer samples in high-density regions, which
appear to lead to an overall larger gradient for faster convergence.
