论文标题
MMSE下限通过Poincaré不平等
An MMSE Lower Bound via Poincaré Inequality
论文作者
论文摘要
本文研究了从噪声观察$ \ Mathbf {y Mathbf {y} \ in \ Mathbb {y Mathbb {r}^k $的噪声（i.e.e。指数家庭的成员。该论文在MMSE上提供了新的下限。为此，首先提出了MMSE的替代表示，这被认为可用于得出MMSE的封闭形式表达式。然后将这种新的表示与庞加莱的不平等一起使用，以在MMSE上提供新的下限。与例如Cramér-rao绑定不同的是，新的界限为输入$ \ Mathbf {x} $上的所有可能发行版。此外，在高斯噪声设置的高噪声状态下，下限在假设$ \ mathbf {x} $是次高斯的假设下都很紧。最后，显示了几个数值示例，这些示例表明该结合在所有噪声方面都表现良好。
This paper studies the minimum mean squared error (MMSE) of estimating $\mathbf{X} \in \mathbb{R}^d$ from the noisy observation $\mathbf{Y} \in \mathbb{R}^k$, under the assumption that the noise (i.e., $\mathbf{Y}|\mathbf{X}$) 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 $\mathbf{X}$. Moreover, the lower bound is shown to be tight in the high-noise regime for the Gaussian noise setting under the assumption that $\mathbf{X}$ is sub-Gaussian. Finally, several numerical examples are shown which demonstrate that the bound performs well in all noise regimes.
