Suppose I have a probability distribution $P$, and suppose that $U$ is the uniform distribution on the same sample space. Then the KL divergence from P to U is
$D_{KL}(U||P) = \sum_x u(x) \log\frac{u(x)}{p(x)}$
We can decompose this as follows:
$D_{KL}(U||P) = \sum_x u(x) \log u(x) - \sum_x u(x) \log p(x)$
The first term is the negative of the entropy of the uniform distribution, or $\log N$ where $N$ is the sample space. The second term is the cross entropy from P to U. So we get
$D_{KL}(U||P) = H(U||P) - H(U) = H(U||P) - \log(N)$
My question: given just the above quantities, is there some way to compute $H(P)$, the entropy of $P$?
Phrased another way: given the KL-divergence from some distribution $P$ to the uniform distribution, is the entropy of $P$ uniquely specified?