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Register a new userComment on the Equality Condition for the I-MMSE Proof of Entropy Power Inequality
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Added by
Alex Dytso
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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The paper establishes the equality condition in the I-MMSE proof of the entropy power inequality (EPI). This is done by establishing an exact expression for the deficit between the two sides of the EPI. Interestingly, a necessary condition for the equality is established by making a connection to the famous Cauchy functional equation.
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Using a sharp version of the reverse Young inequality, and a Renyi entropy comparison result due to Fradelizi, Madiman, and Wang, the authors are able to derive Renyi entropy power inequalities for log-concave random vectors when Renyi parameters belong to $(0,1)$. Furthermore, the estimates are shown to be sharp up to absolute constants.
The matrix version of the entropy-power inequality for real or complex coefficients and variables is proved using a transportation argument that easily settles the equality case. An application to blind source extraction is given.
An extension of the entropy power inequality to the form $N_r^alpha(X+Y) geq N_r^alpha(X) + N_r^alpha(Y)$ with arbitrary independent summands $X$ and $Y$ in $mathbb{R}^n$ is obtained for the Renyi entropy and powers $alpha geq (r+1)/2$.
This paper extends the single crossing point property of the scalar MMSE function, derived by Guo, Shamai and Verdu (first presented in ISIT 2008), to the parallel degraded MIMO scenario. It is shown that the matrix Q(t), which is the difference between the MMSE assuming a Gaussian input and the MMSE assuming an arbitrary input, has, at most, a single crossing point for each of its eigenvalues. Together with the I-MMSE relationship, a fundamental connection between Information Theory and Estimation Theory, this new property is employed to derive results in Information Theory. As a simple application of this property we provide an alternative converse proof for the broadcast channel (BC) capacity region under covariance constraint in this specific setting.
The scalar additive Gaussian noise channel has the single crossing point property between the minimum-mean square error (MMSE) in the estimation of the input given the channel output, assuming a Gaussian input to the channel, and the MMSE assuming an arbitrary input. This paper extends the result to the parallel MIMO additive Gaussian channel in three phases: i) The channel matrix is the identity matrix, and we limit the Gaussian input to a vector of Gaussian i.i.d. elements. The single crossing point property is with respect to the snr (as in the scalar case). ii) The channel matrix is arbitrary, the Gaussian input is limited to an independent Gaussian input. A single crossing point property is derived for each diagonal element of the MMSE matrix. iii) The Gaussian input is allowed to be an arbitrary Gaussian random vector. A single crossing point property is derived for each eigenvalue of the MMSE matrix. These three extensions are then translated to new information theoretic properties on the mutual information, using the fundamental relationship between estimation theory and information theory. The results of the last phase are also translated to a new property of Fishers information. Finally, the applicability of all three extensions on information theoretic problems is demonstrated through: a proof of a special case of Shannons vector EPI, a converse proof of the capacity region of the parallel degraded MIMO broadcast channel (BC) under per-antenna power constrains and under covariance constraints, and a converse proof of the capacity region of the compound parallel degraded MIMO BC under covariance constraint.
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