2
$\begingroup$

Suppose two discrete random variables $X$ and $Y$ defined on finite sets $\mathcal{X}$ and $\mathcal{Y}$ are given and also suppose the conditional distribution $P_{Y|X}$ (i.e, channel) is fixed. We want to look into the following $$f(c):=\min_{P_X: H(X)\geq c}H(Y),$$ which gives us the lower bound on channel's output when the input distribution has entropy lower-bounded by $c$.

If I am not mistaken, as the constraint set is closed and convex, we can simplify the above minimization as $$f(c):=\min_{P_X: H(X)= c}H(Y).$$

The function $f$ can be shown to be convex if both $X$ and $Y$ are binary. However, one can also show that for any $X$ and $Y$ with alphabets $\max\{|\mathcal{X}|, |\mathcal{X}|\}>2$ there exists a channel for which $f(c)$ is not convex. So I want to characterize the lower convex envelope of $f(c)$. As the expectation is a convexification operator, I guess that $$g(c):=\min_{P_{XU}: H(X|U)\geq c}H(Y|U)$$ is the lower convex envelope of $f(c)$ where $U$ is an arbitrary random variable. Is this true? How can one prove (or disprove) this statement?

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.