  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.
