Paper ID sheet UCL-INMA-2012.08


Cramér-Rao bounds for synchronization of rotations

Nicolas Boumal, Amit Singer, P.-A. Absil, Vincent D. Blondel
Synchronization of rotations is the problem of estimating a set of rotations R_i \in SO(n), i=1,...N, based on noisy measurements of relative rotations Ri Rj⊤. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cramér–Rao bounds of synchronization, that is, lower bounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide visualization tools for these bounds and interpretation in terms of random walks in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise.
Key words
synchronization of rotations; estimation on manifolds; estimation on graphs; graph Laplacian; fisher information; Cramér–Rao bounds; distributions on the rotation group; Langevin
Information and Inference (2014) 3 (1): 1-39
BibTeX entry

author = "Nicolas Boumal and Amit Signer and P.-A. Absil and Vincent D. Blondel",
title = "Cramer-Rao bounds for synchronization of rotations", 
fjournal = "Information and Inference",
journal = "Information and Inference",
volume = 3,
number = 1,
year = 2014,
doi = "10.1093/imaiai/iat006",