Empirical Bayes Estimators for Sparse Sequences.

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Srinath, K Pavan 
Venkataramanan, Ramji  ORCID logo  https://orcid.org/0000-0001-7915-5432

The problem of estimating a high-dimensional sparse vector θ ∈ ℝ n from an observation in i.i.d. Gaussian noise is considered. An empirical Bayes shrinkage estimator, derived using a Bernoulli-Gaussian prior, is analyzed and compared with the well-known soft-thresholding estimator using squared-error loss as a measure of performance. We obtain concentration inequalities for the Stein's unbiased risk estimate and the loss function of both estimators. Depending on the underlying θ, either the proposed empirical Bayes (eBayes) estimator or soft-thresholding may have smaller loss. We consider a hybrid estimator that attempts to pick the better of the soft-thresholding estimator and the eBayes estimator by comparing their risk estimates. It is shown that: i) the loss of the hybrid estimator concentrates on the minimum of the losses of the two competing estimators, and ii) the risk of the hybrid estimator is within order 1/√n of the minimum of the two risks. Simulation results are provided to support the theoretical results.

40 Engineering, 46 Information and Computing Sciences, 4006 Communications Engineering, 49 Mathematical Sciences, 4905 Statistics, 4603 Computer Vision and Multimedia Computation
Journal Title
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2018 IEEE International Symposium on Information Theory (ISIT)
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Engineering and Physical Sciences Research Council (EP/N013999/1)
Isaac Newton Trust (1540 (R))