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.
Cite this article
Afonso Bandeira, Jonathan Niles-Weed, Philippe Rigollet, Optimal rates of estimation for multi-reference alignment. Math. Stat. Learn. 2 (2019), no. 1, pp. 25–75DOI 10.4171/MSL/11