# The generalized orthogonal Procrustes problem in the high noise regime

## TL;DR

This paper addresses the challenge of estimating point clouds from noisy observations with unknown rotations in high noise regimes, proposing an invariant polynomial-based method that is statistically optimal and analyzing its variance.

## Contribution

It introduces a simple, efficient invariant polynomial approach for the generalized orthogonal Procrustes problem, effective across different noise levels, and provides new bounds on Cholesky factorization sensitivity.

## Key findings

- Method is statistically optimal in high noise regimes.
- Proposed approach adapts to noise levels for accurate recovery.
- Improved bounds on Cholesky factorization sensitivity are established.

## 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 3-D 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.