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Proceedings of the American Mathematical Society

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Perturbation bounds for eigenspaces under a relative gap condition


Authors: Moritz Jirak and Martin Wahl
Journal: Proc. Amer. Math. Soc. 148 (2020), 479-494
MSC (2010): Primary 15A42, 47A55, 62H25
DOI: https://doi.org/10.1090/proc/14714
Published electronically: September 20, 2019
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Abstract: A basic problem in operator theory is to estimate how a small perturbation affects the eigenspaces of a self-adjoint compact operator. In this paper, we prove upper bounds for the subspace distance, tailored for relative perturbations. As a main example, we consider the empirical covariance operator and show that a sharp bound can be achieved under a relative gap condition. The proof is based on a novel contraction phenomenon, contrasting previous spectral perturbation approaches.


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Additional Information

Moritz Jirak
Affiliation: Institut für Mathematische Stochastik, Technische Universität Braunschweig, Universitätsplatz 2, 38106 Braunschweig, Germany
Email: m.jirak@tu-braunschweig.de

Martin Wahl
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
Email: martin.wahl@math.hu-berlin.de

DOI: https://doi.org/10.1090/proc/14714
Keywords: Relative perturbation bounds, eigenspace, covariance operator
Received by editor(s): May 22, 2018
Received by editor(s) in revised form: December 16, 2018, and May 20, 2019
Published electronically: September 20, 2019
Additional Notes: The research of the second author was partially funded by Deutsche Forschungsgemeinschaft (DFG) via FOR 1735.
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2019 American Mathematical Society