Journal of Scientific Computing

, Volume 79, Issue 2, pp 1135–1160 | Cite as

ALORA: Affine Low-Rank Approximations

  • Alan AyalaEmail author
  • Xavier Claeys
  • Laura Grigori


In this paper we present the concept of affine low-rank approximation for an \(m\times n\) matrix, consisting in fitting its columns into an affine subspace of dimension at most \(k \ll \min (m,n)\). We present the algorithm ALORA that constructs an affine approximation by slightly modifying the application of any low-rank approximation method. We focus on approximations created with the classical QRCP and subspace iteration algorithms. For the former, we discuss existing pivoting techniques and provide a bound for the error when an arbitrary pivoting technique is used. For the case of fsubspace iteration, we prove a result on the convergence of singular vectors, showing a bound that agrees with the one recently proved for the convergence of singular values. Finally, we present numerical experiments using challenging matrices taken from different fields, showing good performance and validating the theoretical framework.


Low rank QR factorization Subspace iteration Affine subspaces 

Mathematics Subject Classification

65F25 65F30 



Funding was provided by H2020 European Research Council (671633) and Agence Nationale de la Recherche (ANR-15-CE23-0017-01).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsINRIA Paris, Sorbonne Université, Univ Paris-Diderot SPC, CNRS, équipe ALPINESParisFrance
  2. 2.Laboratoire Jacques-Louis LionsSorbonne Université, Univ Paris-Diderot SPC, CNRS, INRIA, équipe ALPINESPairsFrance

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