TY - JOUR
T1 - Wasserstein Continuity of Entropy and Outer Bounds for Interference Channels
AU - Polyanskiy, Yury
AU - Wu, Yihong
N1 - Funding Information:
Y. P. was supported in part by the Center for Science of Information, in part by the NSF Science and Technology Center and the NSF CAREER Award within the Division of Computing and Communication Foundations under Grant CCF-09-39370 and Grant CCF-12-53205. Y. Wu was supported in part by NSF within the Division of Computing and Communication Foundations under Grant CCF-14-23088 and Grant CCF-15-27105, in part by NSF within the Division of Information and Intelligent Systems Grant IIS-14-47879, and in part by the Strategic Research Initiative within the College of Engineering at the University of Illinois.
Publisher Copyright:
© 2016 IEEE.
PY - 2016/7
Y1 - 2016/7
N2 - It is shown that under suitable regularity conditions, differential entropy is O(n)-Lipschitz as a function of probability distributions on ℝn with respect to the quadratic Wasserstein distance. Under similar conditions, (discrete) Shannon entropy is shown to be O(n)-Lipschitz in distributions over the product space with respect to Ornstein's d-distance (Wasserstein distance corresponding to the Hamming distance). These results together with Talagrand's and Marton's transportation-information inequalities allow one to replace the unknown multi-user interference with its independent identically distributed approximations. As an application, a new outer bound for the two-user Gaussian interference channel is proved, which, in particular, settles the missing corner point problem of Costa (1985).
AB - It is shown that under suitable regularity conditions, differential entropy is O(n)-Lipschitz as a function of probability distributions on ℝn with respect to the quadratic Wasserstein distance. Under similar conditions, (discrete) Shannon entropy is shown to be O(n)-Lipschitz in distributions over the product space with respect to Ornstein's d-distance (Wasserstein distance corresponding to the Hamming distance). These results together with Talagrand's and Marton's transportation-information inequalities allow one to replace the unknown multi-user interference with its independent identically distributed approximations. As an application, a new outer bound for the two-user Gaussian interference channel is proved, which, in particular, settles the missing corner point problem of Costa (1985).
KW - Entropy
KW - Interference channels
KW - Transport information inequality
KW - Wasserstein distance
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U2 - 10.1109/TIT.2016.2562630
DO - 10.1109/TIT.2016.2562630
M3 - Article
AN - SCOPUS:84976509411
SN - 0018-9448
VL - 62
SP - 3992
EP - 4002
JO - IRE Professional Group on Information Theory
JF - IRE Professional Group on Information Theory
IS - 7
M1 - 7467523
ER -