This paper gives improved Renyi entropy power inequalities (R-EPIs). Consider a sum    $S_{n} = \sum _{k=1}^{n} X_{k}$    of    $n$    independent continuous random vectors taking values on    $ {\mathbb {R}}^{d}$   , and let    $\alpha \in [1, \infty ]$   . An R-EPI provides a lower bound on the order-   $\alpha $    Renyi entropy power of    $S_{n}$    that, up to a multiplicative constant (which may depend in general on    $n, \alpha , d$   ), is equal to the sum of the order-   $\alpha $    Renyi entropy powers of the    $n$    random vectors    $\{X_{k}\}_{k=1}^{n}$   . For    $\alpha =1$   , the R-EPI coincides with the well-known entropy power inequality by Shannon. The first improved R-EPI is obtained by tightening the recent R-EPI by Bobkov and Chistyakov, which relies on the sharpened Young’s inequality. A further improvement of the R-EPI also relies on convex optimization and results on rank-one modification of a real-valued diagonal matrix.
['Eshed Ram', 'Igal Sason']
On Rényi Entropy Power Inequalities