An MMSE Lower Bound via Poincaré Inequality
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This paper studies the minimum mean squared error (MMSE) of estimating 𝐗∈ℝ^d from the noisy observation 𝐘∈ℝ^k, 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.
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