Optimal rates of estimation for multi-reference alignment

Afonso Bandeira, Jonathan Niles-Weed, Philippe Rigollet

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)25-75
Number of pages51
JournalMathematical Statistics and Learning
Issue number1
StatePublished - 2019


  • mixtures of Gaussians
  • Multi-reference alignment
  • orbit retrieval

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Signal Processing
  • Statistics and Probability
  • Theoretical Computer Science


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