A Generalization of the Entropy Power Inequality with Applications

We prove the following generalization of the Entropy Power Inequality: h(Ax) h(A~x) where h(\Delta) denotes (joint-) differential-entropy, x = x 1 : : : xn is a random vector with independent components, ~ x = ~ x 1 : : : ~ xn is a Gaussian vector with independent components such that h(~x i ) = h(x i ), i = 1 : : : n, and A is any matrix. This generalization of the entropy-power inequality is applied to show that a non-Gaussian vector with independent components becomes "closer" to Gaussianity after a linear transformation, where the distance to Gaussianity is measured by the information divergence. Another application is a lower bound, greater than zero, for the mutual-information between non overlapping spectral components of a non-Gaussian white process. Finally, we describe a dual generalization of the Fisher Information Inequality. Key Words: Entropy Power Inequality, Non-Gaussianity, Divergence, Fisher Information Inequality. This research was supported in part by the Wolf..