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Low-rank retractions: a survey and new results

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  • P.-A. Absil
  • I. Oseledets

Abstract

Retractions are a prevalent tool in Riemannian optimization that provides a way to smoothly select a curve on a manifold with given initial position and velocity. We review and propose several retractions on the manifold $${\mathcal {M}}_r$$ M r of rank- $$r$$ r $$m\times n$$ m × n matrices. With the exception of the exponential retraction (for the embedded geometry), which is clearly the least efficient choice, the retractions considered do not differ much in terms of run time and flop count. However, considerable differences are observed according to properties such as domain of definition, boundedness, first/second-order property, and symmetry. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • P.-A. Absil & I. Oseledets, 2015. "Low-rank retractions: a survey and new results," Computational Optimization and Applications, Springer, vol. 62(1), pages 5-29, September.
    • Handle: RePEc:spr:coopap:v:62:y:2015:i:1:p:5-29
    DOI: 10.1007/s10589-014-9714-4
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    References listed on IDEAS

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    1. David G. Luenberger, 1972. "The Gradient Projection Method Along Geodesics," Management Science, INFORMS, vol. 18(11), pages 620-631, July.
    2. P.-A. Absil & Luca Amodei & Gilles Meyer, 2014. "Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries," Computational Statistics, Springer, vol. 29(3), pages 569-590, June.
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    Cited by:

    1. Xiaojing Zhu & Hiroyuki Sato, 2020. "Riemannian conjugate gradient methods with inverse retraction," Computational Optimization and Applications, Springer, vol. 77(3), pages 779-810, December.
    2. K. M. Asim & F. Martínez-Álvarez & A. Basit & T. Iqbal, 2017. "Earthquake magnitude prediction in Hindukush region using machine learning techniques," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 85(1), pages 471-486, January.

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