The generalized orthogonal Procrustes problem in the high noise regime

Abstract

We consider the problem of estimating a cloud of points from numerous noisy observations of that cloud after unknown rotations and possibly reflections. This is an instance of the general problem of estimation under group action, originally inspired by applications in three-dimensional imaging and computer vision. We focus on a regime where the noise level is larger than the magnitude of the signal, so much so that the rotations cannot be estimated reliably. We propose a simple and efficient procedure based on invariant polynomials (effectively: the Gram matrices) to recover the signal, and we assess it against fundamental limits of the problem that we derive. We show our approach adapts to the noise level and is statistically optimal (up to constants) for both the low and high noise regimes. In studying the variance of our estimator, we encounter the question of the sensivity of a type of thin Cholesky factorization, for which we provide an improved bound which may be of independent interest.

Physics-informed dynamic mode decomposition (piDMD)

“…Thus, the solution ( 13) is optimal even when there is noise in both X and Y , as long as the noise has the same distribution in X and Y . There are various problems closely related to the orthogonal Procrustes problem, including the case where A lies on the Steifel manifold (i.e., when A is rectangular with orthonormal columns [48]), there is a weighting matrix [47], missing data [42], or high amounts of noise [46].…”

“…Thus, ( 52) solves the continuous-space shift-invariant piDMD regression (46). Note that the solution (52) depends only on the Fourier coefficients of u and v (50).…”

“…The optimization problem (3) is well-known in the statistical literature as a Procrustes problem [37][38][39][40][41][42][43][44][45][46][47][48][49]. Our recasting of the DMD regression as a Procrustes problem is a new connection, and is the basis of piDMD.…”

Near-Optimal Bounds for Generalized Orthogonal Procrustes Problem via Generalized Power Method

“…The generalized orthogonal Procrustes problem (GOPP) has been studied under many different settings. For its broad applications, we refer the interested readers to [25,24,51,10,39,41,8,19,29,45,48,49] and the reference therein. Our work will mainly focus on the optimization approaches in finding the least squares estimator of the GOPP.…”

“…How to recover the underlying point cloud A and corresponding orthogonal matrices {O i } n i=1 ? This problem, known as the generalized orthogonal Procrustes problem, has found many applications in statistics [24,51], computer vision [10,39,41], and imaging science [8,19,29,45,48,49].…”