If we know it’s a linear relationship: given correlation, var of x, var of y, we can deduce that $y=\beta x+\epsilon$.
Now, if we know it’s a nonlinear relationship: given correlation information (say, mutual information), can we conduct similar deductions in the context of nonlinear?
Unfortunately, no. We can’t deduce y based on x and multual information.
If linear is a male elephant, given the knowledge that this creature has two huge ears, we can safely deduce that it also has a long nose. However, nonlinear is not a female elephant, instead it is the whole animal world. It can be a pink pig with two big ears but a short nose, or 大耳朵图图 who has only a hat between his eyes.