Oracle Cramér-Rao Bounds for Sparse Vector Estimation

Abstract

In this paper, we revisit the Cramér-Rao bound (CRB) for deterministic sparse vector estimation under a general observation model. For a given support set <inline-formula><tex-math notation="LaTeX">$\mathcal {S}$</tex-math></inline-formula>, we identify and analyze two alternative oracle CRB formulations: (i) a <italic>reduced oracle CRB</italic>, <inline-formula><tex-math notation="LaTeX">${\mathbf{B}}_{1}$</tex-math></inline-formula>, constructed from the <inline-formula><tex-math notation="LaTeX">$|\mathcal {S}|\times |\mathcal {S}|$</tex-math></inline-formula> Fisher information matrix (FIM) restricted to the oracle subspace, and (ii) a <italic>projected CRB</italic>, <inline-formula><tex-math notation="LaTeX">${\mathbf{B}}_{2}$</tex-math></inline-formula>, obtained by projecting the inverse full-dimensional FIM onto the support subspace. We prove that, whenever both bounds exist, the CRBs satisfy <inline-formula><tex-math notation="LaTeX">${\mathbf{B}}_{2} \succeq {\mathbf{B}}_{1}$</tex-math></inline-formula>. Moreover, we derive a closed-form expression for the gap between the bounds in terms of the cross-information blocks of the FIM, along with an explicit equality condition. We further establish a unifying interpretation of the bounds through the lens of general reparametrization theory. Numerical simulations validate the established order relation between the bounds and demonstrate the relations between the bounds and sparse vector estimators in linear Gaussian models.