Intrinsic Cramer-Rao bounds and subspace estimation accuracy

Signal processing estimation problems are traditionally posed for a set of given, if unknown, parameters, such as angle and/or Doppler. Nevertheless, there are estimation problems on manifolds where no set of intrinsic coordinates exist. One example encountered frequently is the problem of estimating a particular subspace. The set of subspaces, called the Grassmann manifold, has no fixed coordinate system associated with it. This paper addresses the problem of applying classical Cramer-Rao analysis to determine tire fundamental bounds of estimation accuracy on arbitrary manifolds. Coordinate-free versions of the Cramer-Rao bound are derived to accomplish this. These bounds are then applied to the specific problem of estimating the subspace given an independent collection of data snapshots. The root-mean-square-error of the standard method of estimating subspaces using singular valve decomposition is compared to the intrinsic Cramer-Rao bound by varying both the SNR of the unknown subspace and the sample support. It will be seen that this SVD-based method yields accuracies very close to Cramer-Rao bound, establishing that the principal invariant subspace provides an excellent estimator of an unknown subspace, a conclusion that would not in general be possible without coordinate-free Cramer-Rao bounds.