Minoration via mixed volumes and Cover’s problem for general channels

Liu, Jingbo

doi:10.1007/s00440-022-01111-6

Cited by 4 publications

(1 citation statement)

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“…The similarities between entropic and geometric inequalities are well-known (e.g. [19][4]) and sometimes can be employed to prove new results in network information theory [34]. Intuitively, differential entropy can be seen as analogous to the logarithmic volume of a set, and the sum of random variables is analogous to the Minkowski sum of sets.…”

Section: A Geometric Inequality Analogous To Lemmamentioning

confidence: 99%

Stability of the Gaussian Stationary Point in the Han–Kobayashi Region for Z-Interference Channels

Liu

2023

IEEE Trans. Inform. Theory

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The Gaussian stationary point in an inequality motivated by the Z-interference channel was recently conjectured by Costa, Nair, Ng, and Wang to be the global optimizer, which, if true, would imply the optimality of the Han-Kobayashi region for the Gaussian Z-interference channel. This conjecture was known to be true for some parameter regimes, but the validity for all parameters, although suggested by Gaussian tensorization, was previously open. In this paper we construct several counterexamples showing that this conjecture may fail in certain regimes: A simple construction without Hermite polynomial perturbation is proposed, where distributions far from Gaussian are analytically shown to be better than the Gaussian stationary point. As alternatives, we consider perturbation along geodesics under either the L 2 or Wasserstein-2 metric, showing that the Gaussian stationary point is unstable in a certain regime. Similarity to stability of the Levy-Cramer theorem is discussed. The stability phase transition point admits a simple characterization in terms of the maximum eigenvalue of the Gaussian maximizer. Similar to the Holley-Stroock principle, we can show that in the stable regime the Gaussian stationary point is optimal in a neighborhood under the L ∞ -norm with respect to the Gaussian measure. Allowing variable power control, we show that the Gaussian optimizers for the Han-Kobayashi region always lie in the stable regime. Finally, an amended conjecture is proposed.

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Section: A Geometric Inequality Analogous To Lemmamentioning

confidence: 99%