  We consider minimum variance estimation within the sparse linear Gaussian
model (SLGM). A sparse vector is to be estimated from a linearly transformed
version embedded in Gaussian noise. Our analysis is based on the theory of
reproducing kernel Hilbert spaces (RKHS). After a characterization of the RKHS
associated with the SLGM, we derive novel lower bounds on the minimum variance
achievable by estimators with a prescribed bias function. This includes the
important case of unbiased estimation. The variance bounds are obtained via an
orthogonal projection of the prescribed mean function onto a subspace of the
RKHS associated with the SLGM. Furthermore, we specialize our bounds to
compressed sensing measurement matrices and express them in terms of the
restricted isometry and coherence parameters. For the special case of the SLGM
given by the sparse signal in noise model (SSNM), we derive closed-form
expressions of the minimum achievable variance (Barankin bound) and the
corresponding locally minimum variance estimator. We also analyze the effects
of exact and approximate sparsity information and show that the minimum
achievable variance for exact sparsity is not a limiting case of that for
approximate sparsity. Finally, we compare our bounds with the variance of three
well-known estimators, namely, the maximum-likelihood estimator, the
hard-thresholding estimator, and compressive reconstruction using the
orthogonal matching pursuit.
