Monte Carlo extrinsic estimators of manifold-valued parameters

Monte Carlo (MC) methods have become an important tool for inferences in non-Gaussian and non-Euclidean settings. We study their use in those signal/image processing scenarios where the parameter spaces are certain Riemannian manifolds (finite-dimensional Lie groups and their quotient sets). We investigate the estimation of means and variances of the manifold-valued parameters, using two popular sampling methods: independent and importance sampling. Using Euclidean embeddings, we specify a notion of extrinsic means, employ Monte Carlo methods to estimate these means, and utilize large-sample asymptotics to approximate the estimator covariances. Experimental results are presented for target pose estimation (orthogonal groups) and signal subspace estimation (Grassmann manifolds). Asymptotic covariances are utilized to construct confidence regions, to compare estimators, and to determine the sample size for MC sampling.