  In this paper, we establish optimal rates of adaptive estimation of a vector
in the multi-reference alignment model, a problem with important applications
in fields such as signal processing, image processing, and computer vision,
among others. We describe how this model can be viewed as a multivariate
Gaussian mixture model under the constraint that the centers belong to the
orbit of a group. This enables us to derive matching upper and lower bounds
that feature an interesting dependence on the signal-to-noise ratio of the
model. Both upper and lower bounds are articulated around a tight local control
of Kullback-Leibler divergences that showcases the central role of moment
tensors in this problem.
