Bound on mutual information after nonlinearity

I can calculate the mutual information between two, (possibly multivariate) Gaussian random variables $I[X,Y]$. Is there anything I can say about the mutual information (perhaps establish a bound) between $I[X, f(Y)]$, where $f$ is some nonlinearity? I understand the the data processing inequality establishes $I[X,Y]$ as an upper bound, but is there a stronger statement I can make given that $X$ and $Y$ are Gaussian and if the nonlinearity is sufficiently "simple" (for example, a squared nonlinearity)?

If $f$ is a 1-1 function then $f(Y)$ gives you the same info as $Y$ and so $I(X;Y)=I(X; f(Y))$. Formally, since you can invert $f$, we have by multiple uses of the data processing inequality $$I(X;Y)\geq I(X;f(Y)) \geq I(X;f^{-1}(f(Y))) = I(X;Y)$$
• How about if $f$ is no longer one-to-one (such as a thresholded function, or a squared nonlinearity)? – user2452976 Nov 9 '17 at 19:58
• Another case is when $X,Y$ are independent, then $X, f(Y)$ are also independent (regardless of the function $f$, even if $f$ is not 1-1) so $I(X;Y)=I(X;f(Y))=0$. – Michael Nov 9 '17 at 20:00