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Journal Articles Journal of Scientific Computing Year : 2019

ALORA: Affine Low-Rank Approximations

ALORA: Approximations affines de rang faible


In this paper we present the concept of affine low-rank approximation for an m x n matrix, consisting in fitting its columns into an affine subspace of dimension at most k << min(m, n). We show that the optimal affine approximation can be obtained by applying an orthogonal projection to the matrix before constructing its best approximation. Moreover, we present the algorithm ALORA that constructs an affine approximation by slightly modifying the application of any lowrank approximation method. We focus on approximations created with the classical QRCP and subspace iteration algorithms. For the former, we present a detailed analysis of the existing pivoting techniques and furthermore, we provide a bound for the error when an arbitrary pivoting technique is used. For the case of subspace iteration, we prove a result on the convergence of singular vectors, showing a bound that is in agreement with the one for convergence of singular values proved recently. Finally, we present numerical experiences using challenging matrices taken from different fields, showing good performance and validating the theoretical framework.
Dans cet article, nous présentons le concept d’approximation affine de rang faible pour des matrices rectangulaires. Nous montrons comment construire ce type d’approximation en utilisant des projection orthogonaux avec des factorisations QR et itération sur sous-espaces. Nous proposons un algorithme (ALORA) pour calculer une approximation affine de rang faible et le comparons avec des méthodes classiques. Des expériences numériques avec des matrices provenant de différents champs intéressants montrent des bonnes performances et valident le cadre théorique.
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Dates and versions

hal-01762882 , version 1 (10-04-2018)


  • HAL Id : hal-01762882 , version 1


Alan Ayala, Xavier Claeys, Laura Grigori. ALORA: Affine Low-Rank Approximations. Journal of Scientific Computing, 2019, 79 (2), pp.1135-1160. ⟨hal-01762882⟩
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