Learning rates for Gaussian mixtures under group action

Victor-Emmanuel Brunel
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:471-491, 2019.

Abstract

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

Cite this Paper

BibTeX
@InProceedings{pmlr-v99-brunel19a, title = {Learning rates for Gaussian mixtures under group action}, author = {Brunel, Victor-Emmanuel}, booktitle = {Proceedings of the Thirty-Second Conference on Learning Theory}, pages = {471--491}, year = {2019}, editor = {Beygelzimer, Alina and Hsu, Daniel}, volume = {99}, series = {Proceedings of Machine Learning Research}, month = {25--28 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v99/brunel19a/brunel19a.pdf}, url = {https://proceedings.mlr.press/v99/brunel19a.html}, abstract = {We study the pointwise maximum likelihood estimation rates for a class of Gaussian mixtures that are invariant under the action of some isometry group. This model is also known as multi-reference alignment, where random rotations of a given vector are observed, up to Gaussian noise. We completely characterize the speed of the maximum likelihood estimator, by giving a comprehensive description of the likelihood geometry of the model. We show that the unknown parameter can always be decomposed into two components, one of which can be estimated at the fast rate $n^{-1/2}$, the other one being estimated at the slower rate $n^{-1/4}$. We provide an algebraic description and a geometric interpretation of these facts.} }
Endnote
%0 Conference Paper %T Learning rates for Gaussian mixtures under group action %A Victor-Emmanuel Brunel %B Proceedings of the Thirty-Second Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2019 %E Alina Beygelzimer %E Daniel Hsu %F pmlr-v99-brunel19a %I PMLR %P 471--491 %U https://proceedings.mlr.press/v99/brunel19a.html %V 99 %X We study the pointwise maximum likelihood estimation rates for a class of Gaussian mixtures that are invariant under the action of some isometry group. This model is also known as multi-reference alignment, where random rotations of a given vector are observed, up to Gaussian noise. We completely characterize the speed of the maximum likelihood estimator, by giving a comprehensive description of the likelihood geometry of the model. We show that the unknown parameter can always be decomposed into two components, one of which can be estimated at the fast rate $n^{-1/2}$, the other one being estimated at the slower rate $n^{-1/4}$. We provide an algebraic description and a geometric interpretation of these facts.
APA
Brunel, V.. (2019). Learning rates for Gaussian mixtures under group action. Proceedings of the Thirty-Second Conference on Learning Theory, in Proceedings of Machine Learning Research 99:471-491 Available from https://proceedings.mlr.press/v99/brunel19a.html.

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