# Does a 1-Lipschitz function preserve mutual information between two random variables?

Suppose we have a 1-Lipschitz function $$f$$ such that 1-Lipschitzness is preserved, with $$D_A(f(X), f(Y)) \leq D_B(X, Y)$$ for some metric spaces $$A$$ and $$B$$.

Does this also imply that $$I(f(X); f(Y)) = I(X;Y)$$? It seems intuitive that this would be the case if the 1-lipschitz function is injective, but I don't know if this is true, so that the mutual information is preserved under such invertible transformation. Is there a way to prove this?

No, it does not. Let $$X$$ and $$Y$$ be non-independent $$A$$-valued random variables, and let $$f:A\to B$$ be a constant function. Then $$I(f(X);f(Y))=0.
• Note the OP assumed $f$ is injective Oct 31 '19 at 3:17
• Sorry I should have made it clearer: I don't know if $f$ is injective or not, but from @e.lipnowski comments it seems that $f$ isn't injective even if it's 1-Lipschitz, in which case I should suppose the MI is not preserved? Oct 31 '19 at 11:08