On Intrinsic Cramér-Rao Bounds for Riemannian Submanifolds and Quotient Manifolds

We study Cramér-Rao bounds (CRB's) for estimation problems on Riemannian manifolds. In [S. T. Smith, “Covariance, Subspace, and Intrinsic Cramér-Rao bounds,” IEEE Trans. Signal Process., vol. 53, no. 5, 1610–1630, 2005], the author gives intrinsic CRB's in the form of matrix inequalities relating the covariance of estimators and the Fisher information of estimation problems. We focus on estimation problems whose parameter space <formula formulatype="inline"><tex Notation="TeX">$\bar{\cal P}$</tex></formula> is a Riemannian submanifold or a Riemannian quotient manifold of a parent space <formula formulatype="inline"><tex Notation="TeX">${\cal P}$</tex></formula>, that is, estimation problems on manifolds with either deterministic constraints or ambiguities. The CRB's in the aforementioned reference would be expressed w.r.t. bases of the tangent spaces to <formula formulatype="inline"><tex Notation="TeX">$\bar{\cal P}$</tex></formula>. In some cases though, it is more convenient to express covariance and Fisher information w.r.t. bases of the tangent spaces to <formula formulatype="inline"><tex Notation="TeX">${\cal P}$</tex></formula>. We give CRB's w.r.t. such bases expressed in terms of the geodesic distances on the parameter space. The bounds are valid even for singular Fisher information matrices. In two examples, we show how the CRB's for synchronization problems (including a type of sensor network localization problem) differ in the presence or absence of anchors, leading to bounds for estimation on either submanifolds or quotient manifolds with very different interpretations.

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