Partially Linear Bayesian Estimation with Application to Sparse Approximations
Tomer Michaeli, Daniel Sigalov, Yonina C. Eldar · 2011
We address the problem of estimating a random vector X from two sets of measurements Y and Z, such that the estimator is linear in Y. We show that the partially linear minimum mean squared error (PLMMSE) estimator does not require knowing the joint distribution of X and Y in full, but rather only its second-order moments. This renders it of potential interest in various applications. We further show that the PLMMSE method is minimax-optimal among all estimators that solely depend on the second-order statistics of X and Y. Finally, we demonstrate our approach in the context of recovering a vector, which is sparse in a unitary dictionary, from two sets of noisy measurements. We show that in this setting PLMMSE estimation has a clear computational advantage, while its performance is comparable to state-of-the-art algorithms.