Uncertainty Quantification of Spectral Estimator and MLE for Orthogonal Group Synchronization
Ziliang Samuel Zhong, Shuyang Ling
TL;DR
This work develops a precise uncertainty quantification framework for orthogonal group synchronization under Gaussian noise by deriving a second-order, tangent-space expansion for both the maximum likelihood estimator and the spectral estimator. The leading stochastic term is an antisymmetric Gaussian on the tangent space, enabling near-optimal min-max risk bounds and construction of confidence regions for each group element and pairwise products. The results are complemented by proofs sketching a one-step Newton approach on manifolds and extensive numerics that confirm normal-like behavior and the predicted second-order terms. The framework offers a general approach to approximate distributions of estimators under manifold constraints and has potential extensions to sub-Gaussian noise and other synchronization problems.
Abstract
Orthogonal group synchronization aims to recover orthogonal group elements from their noisy pairwise measurements. It has found numerous applications including computer vision, imaging science, and community detection. Due to the orthogonal constraints, it is often challenging to find the least squares estimator in presence of noise. In the recent years, semidefinite relaxation (SDR) and spectral methods have proven to be powerful tools in recovering the group elements. In particular, under additive Gaussian noise, the SDR exactly produces the maximum likelihood estimator (MLE), and both MLE and spectral methods are able to achieve near-optimal statistical error. In this work, we take one step further to quantify the uncertainty of the MLE and spectral estimators by considering their distributions. By leveraging the orthogonality constraints in the likelihood function, we obtain a second-order expansion of the MLE and spectral estimator with the leading terms as an anti-symmetric Gaussian random matrix that is on the tangent space of the orthogonal matrix. This also implies state-of-the-art min-max risk bounds and a confidence region of each group element as a by-product. Our works provide a general theoretical framework that is potentially useful to find an approximate distribution of the estimators arising from many statistical inference problems with manifold constraints. The numerical experiments confirm our theoretical contribution.