Intrinsic Variance Lower Bound (IVLB): An Extension of the Cramer-Rao Bound to Riemannian Manifolds

Abstract

We consider parametric statistical models in which the parameter space Θ is a connected Riemannian manifold. This mathematical structure on the parameter space subsumes, as special cases, submanifolds of Euclidean spaces appearing in parametric estimation scenarios with a priori smooth deterministic constraints, and quotient spaces (such as Grassmann manifolds) which arise in certain parametric estimation scenarios with ambiguities. The Riemannian structure on the parameter space Θ turns it into a metric space and the associated Riemannian distance is used here to quantify estimation errors. We present the Intrinsic Variance Lower Bound (IVLB) which places a lower limit on the accuracy, measured in terms of the mean-square Riemannian distance, of unbiased estimators taking values in Θ. The IVLB depends both on the curvature of the parameter space and a coordinate-free extension of the well-known Fisher information matrix (FIM). We show that for flat Euclidean spaces, the IVLB collapses to the Cramér-Rao Bound (CRB). In this sense, we may interpret the IVLB as a generalization of the CRB for curved parameter spaces. Computer simulations illustrating the application of the IVLB are included.

A dynamic estimator on SE(3) using range-only measurements

“…The problem of simultaneous range-only attitude and position estimation was addressed in a Maximum likelihood framework in [9] by formulating a constrained optimization problem on the Special Euclidean group SE(3) and resorting to generalized intrinsic gradient and Newton algorithms. The performance of the derived estimator is very close to the theoretical bounds provided by the Intrinsic Variance Lower Bound IVLB [10]. However, no convergence warranties were given and simulations revealed the existence of local minima.…”

“…Assuming a static rigid body (Ṙ = 0 andṗ = 0), the error dynamics can 47th IEEE CDC, Cancun, Mexico, Dec. [9][10][11]2008 WeA13.3 be written from (5)-(6) as…”

“…Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. [9][10][11]2008 WeA13.3 978-1-4244-3124-3/08/$25.00 ©2008 IEEE Using a suitable Lyapunov candidate function based on the range observations, and under certain conditions, local asymptotical convergence of the position and attitude estimates to the true values is proven. The conditions required to obtain this result consist in having at least a set of noncoplanar landmarks and placing the origin of the body reference frame at the centroid of the beacons.…”

Trust-Region Methods on Riemannian Manifolds

“…Applications appear in various areas, including computer vision [MKS01], machine learning [NA05], maximum likelihood estimation [Smi05], [XB05], electronic structure computation [LE00], system balancing [HM94], model reduction [YL99], and robot manipulation [HHM02].…”

Gauss–Newton method for convex composite optimizations on Riemannian manifolds

“…Applications appear in various areas, including computer vision [31], machine learning [34], maximum likelihood estimation [45,56], electronic structure computation [30], system balancing [19], model reduction [57], and robot manipulation [18]. A lot of algorithms such as steepest descent method, trust-region method, conjugate gradient method and so on have been extended to solve optimization problems on Riemannian manifolds (see, e.g., [1,23,46,47] and the references therein).…”