An MMSE Lower Bound via Poincaré Inequality

Ian Zieder, Alex Dytso, Martina Cardone

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

This paper studies the minimum mean squared error (MMSE) of estimating X ∈ ℝd from the noisy observation Y ∈ ℝk, under the assumption that the noise (i.e., Y|X) is a member of the exponential family. The paper provides a new lower bound on the MMSE. Towards this end, an alternative representation of the MMSE is first presented, which is argued to be useful in deriving closed-form expressions for the MMSE. This new representation is then used together with the Poincaré inequality to provide a new lower bound on the MMSE. Unlike, for example, the Cramér-Rao bound, the new bound holds for all possible distributions on the input X. Moreover, the lower bound is shown to be tight in the high-noise regime for the Gaussian noise setting under the assumption that X is sub-Gaussian. Finally, several numerical examples are shown which demonstrate that the bound performs well in all noise regimes.

Original languageEnglish (US)
Title of host publication2022 IEEE International Symposium on Information Theory, ISIT 2022
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages957-962
Number of pages6
ISBN (Electronic)9781665421591
DOIs
StatePublished - 2022
Event2022 IEEE International Symposium on Information Theory, ISIT 2022 - Espoo, Finland
Duration: Jun 26 2022Jul 1 2022

Publication series

Name2022 IEEE International Symposium on Information Theory (ISIT)

Conference

Conference2022 IEEE International Symposium on Information Theory, ISIT 2022
Country/TerritoryFinland
CityEspoo
Period6/26/227/1/22

Bibliographical note

Publisher Copyright:
© 2022 IEEE.

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