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We address the problem of estimating a random vector from two sets of measurements and , such that the estimator is linear in . We show that the partially linear minimum mean-square error (PLMMSE) estimator does not require knowing the joint distribution of and 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 and . We demonstrate our approach in the context of recovering a signal, which is sparse in a unitary dictionary, from noisy observations of it and of a filtered version. We show that in this setting PLMMSE estimation has a clear computational advantage, while its performance is comparable to state-of-the-art algorithms. We apply our approach both in static and in dynamic estimation applications. In the former category, we treat the problem of image enhancement from blurred/noisy image pairs. We show that PLMMSE estimation performs only slightly worse than state-of-the art algorithms, while running an order of magnitude faster. In the dynamic setting, we provide a recursive implementation of the estimator and demonstrate its utility in tracking maneuvering targets from position and acceleration measurements.