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A lower bound on the differential entropy of log-concave random vectors with applications. (arXiv:1704.07766v1 [cs.IT])

来源于:arXiv
We derive a lower bound on the differential entropy of a log-concave random variable $X$ in terms of the $p$-th absolute moment of $X$. The new bound leads to a reverse entropy power inequality with an explicit constant, and to new bounds on the rate-distortion function and the channel capacity. Specifically, we study the rate-distortion function for log-concave sources and distortion measure $| x - \hat x|^r$, and we establish that the difference between the rate distortion function and the Shannon lower bound is at most $\log(2 \sqrt{\pi e}) \approx 2.5$ bits, independently of $r$ and the target distortion $d$. For mean-square error distortion, the difference is at most $\log (\sqrt{2 \pi e}) \approx 2$ bits, regardless of $d$. The bounds can be further strengthened if the source, in addition to being log-concave, is symmetric. In particular, we establish that for mean-square error distortion, the difference is at most $\log (\sqrt{\pi e}) \approx 1.5$ bits, regardless of $d$. We als 查看全文>>