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We study the pointwise maximum likelihood estimation rates for a class ofGaussian mixtures that are invariant under the action of some isometry group.This model is also known as multi-reference alignment, where random isometriesof a given vector are observed, up to Gaussian noise. We completelycharacterize the speed of the maximum likelihood estimator, by giving acomprehensive description of the likelihood geometry of the model. We show thatthe unknown parameter can always be decomposed into two components, one ofwhich can be estimated at the fast rate $n^{-1/2}$, the other one beingestimated at the slower rate $n^{-1/4}$. We provide an algebraic descriptionand a geometric interpretation of these facts.

Learning rates for Gaussian mixtures under group invariance