A mutual information equality

Question

$x$ and $y$ are two random variables and $I(u;v)$ is the mutual information between random variables $u$ and $v$. Does the following equality hold? $$\text{argmax}_a I(y;ax)=\text{argmin}_a I(y;y-ax).$$

Answer 1

It can be seen that for any $a\neq 0$, $I(y;ax)=I(y;x)$ because $y\rightarrow ax\rightarrow x$ and $y\rightarrow x\rightarrow ax$ are both Markov chain hence by the data processing inequality both $I(y;x)\leq I(y;ax)$ and $I(y;x)\geq I(y;ax)$ respectively.

Since any $a\neq 0$ is a maximizer of $I(y;ax)$, your question is ill defined.