L1 Estimation in Gaussian Noise: On the Optimality of Linear Estimators
2023 IEEE International Symposium on Information Theory (ISIT)(2023)
摘要
Consider the problem of estimating a random variable X in Gaussian noise under L
1
fidelity criteria. It is well-known that in the L
1
setting, the optimal Bayesian estimator is given by the conditional median. The goal of this work is to characterize the set of prior distributions on X for which the conditional median corresponds to a linear estimator. This work shows that neither discrete nor compactly supported distributions can induce a linear conditional median. Moreover, under certain non-trivial restrictions on the set of allowed probability distributions, the Gaussian is shown to be the only solution that induces a linear conditional median.
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关键词
conditional median,Gaussian noise,L1 estimation,L1 fidelity criteria,linear conditional median,linear estimator optimality,optimal Bayesian estimator,prior distributions,probability distributions
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