Suppose I have Markov chains: $$X \rightarrow f(X) \rightarrow g(X)$$ $$Y \rightarrow f(Y) \rightarrow g(Y)$$
where it is known that minimizing the $\mathbb{E}(g(X)) - \mathbb{E}(g(Y))$ minimizes the Wasserstein distance between X and Y, and that $f$ and $g$ are 1-Lipschitz, and $g = h \circ f$ where $h$ is also 1-Lipschitz.
Firstly, suppose now I maximize the mutual informations $I(f(X); g(Y))$ and $I(f(Y); g(X))$ through some MI estimator, does this show that $I(f(X); f(Y))$ and $I(g(X); g(Y))$ is also maximized?
If so, does the maximization of $I(g(X); g(Y))$ lead to a decrease in the Wasserstein distance between $X$ and $Y$?
It seems intuitive that maximizing the MI between 2 random variables also minimize their wasserstein distance. However, in this case I am minimizing the processed version of the random variables down the markov chain, and not the variables themselves.
Any help would be greatly appreciated.
