A lower bound on the estimator variance for the sparse linear model

We study the performance of estimators of a sparse nonrandom vector based on an observation which is linearly transformed and corrupted by white Gaussian noise. Using the framework of reproducing kernel Hilbert spaces, we derive a new lower bound on the estimator variance for a given differentiable bias function (including the unbiased case) and an almost arbitrary transformation matrix (including the underdetermined case considered in compressed sensing theory). For the special case of a sparse vector corrupted by white Gaussian noise-i.e., without a linear transformation-and unbiased estimation, our lower bound improves on a previously proposed bound.