Cramér-Rao bounds for synchronization of rotations

Abstract

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 R_i R_j^T. 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 interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise.

References in corpus (6)

• How to generate random matrices from the classical compact groups
• Spectral measure of large random Hankel, Markov and Toeplitz matrices
• Random Walks and Electric Networks
• Estimation and Registration on Graphs
• Exact and Stable Recovery of Rotations for Robust Synchronization
• A Cheeger Inequality for the Graph Connection Laplacian

Cited by in corpus (24)

• Nonconvex phase synchronization
• Tightness of the maximum likelihood semidefinite relaxation for angular synchronization
• Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models
• Robust Rotation Synchronization via Low-rank and Sparse Matrix Decomposition
• Optimality and Sub-optimality of PCA for Spiked Random Matrices and Synchronization
• Non-Bayesian Estimation Framework for Signal Recovery on Graphs
• The Geometry of Synchronization Problems and Learning Group Actions
• Deformed Laplacians and spectral ranking in directed networks
• Concentration of the Kirchhoff index for Erdos-Renyi graphs
• Exact and Stable Recovery of Rotations for Robust Synchronization
• A Cheeger Inequality for the Graph Connection Laplacian
• Stochastic Primal-Dual Method on Riemannian Manifolds with Bounded Sectional Curvature
• Representation Theoretic Patterns in Multi-Frequency Class Averaging for Three-Dimensional Cryo-Electron Microscopy
• 3-D Relative Localization for Multi-Robot Systems with Angle and Self-Displacement Measurements
• Tightness of the semidefinite relaxation for orthogonal trace-sum maximization
• Lagrangian Duality in 3D SLAM: Verification Techniques and Optimal Solutions
• On the Estimation Performance and Convergence Rate of the Generalized Power Method for Phase Synchronization
• Advances in Inference and Representation for Simultaneous Localization and Mapping
• VICAN: Very Efficient Calibration Algorithm for Large Camera Networks
• On the Whitney distortion extension problem for $C^m({\mathbb{R}}^n)$ and $C^{\infty }({\mathbb{R}}^n)$ and its applications to interpolation and alignment of data in ${\mathbb{R}}^n$
• On the Whitney Extension-Interpolation-Alignment problem for almost isometries with small distortion in ${\mathbb{R}}^D$
• Parameter Estimation on Homogeneous Spaces
• An intrinsic Cramér-Rao bound on Lie groups
• Rotation Averaging in a Split Second: A Primal-Dual Method and a Closed-Form for Cycle Graphs