Nicolas Boumal

Cramér-Rao bounds for synchronization of rotations

Synchronization of rotations is the problem of estimating a set of rotations Ri 2 SO(n);i = 1:::N based on noisy measurements of relative rotations RiR > . 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 Cram er-Rao bounds of synchronization, that is, lower-bounds on the variance of unbiased estimators. We nd 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. Synchronization of rotations, estimation on manifolds, estimation on graphs, graph Laplacian, Fisher information, Cram er-Rao bounds, distributions on the rotation group, Langevin. 2000 Math Subject Classication: 62F99, 94C15, 22C05, 05C12,

Nonconvex Phase Synchronization

We estimate

Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

n

Global rates of convergence for nonconvex optimization on manifolds

phases (angles) from noisy pairwise relative phase measurements. The task is modeled as a nonconvex least-squares optimization problem. It was recently shown that this problem can be solved in polynomial time via convex relaxation, under some conditions on the noise. In this paper, under similar but more restrictive conditions, we show that a modified version of the power method converges to the global optimum. This is simpler and (empirically) faster than convex approaches. Empirically, they both succeed in the same regime. Further analysis shows that, in the same noise regime as previously studied, second-order necessary optimality conditions for this quadratically constrained quadratic program are also sufficient, despite nonconvexity.

A discrete regression method on manifolds and its application to data on SO(n)

Maximum likelihood estimation problems are, in general, intractable optimization problems. As a result, it is common to approximate the maximum likelihood estimator (MLE) using convex relaxations. In some cases, the relaxation is tight: it recovers the true MLE. Most tightness proofs only apply to situations where the MLE exactly recovers a planted solution (known to the analyst). It is then sufficient to establish that the optimality conditions hold at the planted signal. In this paper, we study an estimation problem (angular synchronization) for which the MLE is not a simple function of the planted solution, yet for which the convex relaxation is tight. To establish tightness in this context, the proof is less direct because the point at which to verify optimality conditions is not known explicitly. Angular synchronization consists in estimating a collection of n phases, given noisy measurements of the pairwise relative phases. The MLE for angular synchronization is the solution of a (hard) non-bipartite Grothendieck problem over the complex numbers. We consider a stochastic model for the data: a planted signal (that is, a ground truth set of phases) is corrupted with non-adversarial random noise. Even though the MLE does not coincide with the planted signal, we show that the classical semidefinite relaxation for it is tight, with high probability. This holds even for high levels of noise.

Near-Optimal Bounds for Phase Synchronization

We consider the minimization of a cost function

Bispectrum Inversion With Application to Multireference Alignment

f

A Riemannian subgradient algorithm for economic dispatch with valve-point effect

on a manifold

Interpolation and Regression of Rotation Matrices

M

Estimation of a Multi-fascicle Model from Single B-Value Data with a Population-Informed Prior

using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance