August 2022 Lower bounds for invariant statistical models with applications to principal component analysis
Martin Wahl
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 58(3): 1565-1589 (August 2022). DOI: 10.1214/21-AIHP1193


This paper develops nonasymptotic information inequalities for the estimation of the eigenspaces of a covariance operator. These results generalize previous lower bounds for the spiked covariance model, and they show that recent upper bounds for models with decaying eigenvalues are sharp. The proof relies on lower bound techniques based on group invariance arguments. These techniques can also be applied to a variety of other statistical models.

Ce travail établit des inégalités d’information non-asymptotiques dans le cadre de l’estimation des espaces propres d’un opérateur de covariance. Ces résultats généralisent d’une part des minorations antérieures valables pour le modèle de perturbation de la covariance, et montrent d’autre part l’optimalité de majorations récentes établies dans des modèles sous contraintes de décroissance des valeurs propres. La preuve repose sur des nouvelles techniques de minoration basées sur des arguments d’invariance de groupes. Ces techniques peuvent également être appliquées à une variété d’autres modèles statistiques.


The author would like to thank Markus Reiß, Holger Kösters and the two anonymous referees for their helpful comments and remarks and Alain Rouault for very helpful discussions and comments.


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Martin Wahl. "Lower bounds for invariant statistical models with applications to principal component analysis." Ann. Inst. H. Poincaré Probab. Statist. 58 (3) 1565 - 1589, August 2022.


Received: 25 September 2020; Revised: 25 March 2021; Accepted: 20 May 2021; Published: August 2022
First available in Project Euclid: 14 July 2022

MathSciNet: MR4452643
zbMATH: 1493.62025
Digital Object Identifier: 10.1214/21-AIHP1193

Primary: 62H25
Secondary: 60B20 , 62B10

Keywords: Covariance operator , Equivariant model , Fisher information , large deviations , lower bounds , principal components , special orthogonal group , van Trees inequality

Rights: Copyright © 2022 Association des Publications de l’Institut Henri Poincaré


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Vol.58 • No. 3 • August 2022
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