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On the Accuracy of Cross and Column Low-Rank Maxvol Approximations in Average

  • GENERAL NUMERICAL METHODS
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Abstract

The article considers the problem of low-rank column and cross (\(CGR\), \(CUR\)) approximation of matrices in the Frobenius norm up to a fixed factor \(1 + \varepsilon \). It is proved that, for random matrices, in average, an estimate of the form \(1 + \varepsilon \leqslant \tfrac{{m + 1}}{{m - r + 1}}\tfrac{{n + 1}}{{n - r + 1}}\) holds, where \(m\) and \(n\) are the number of rows and columns of the cross approximation. Thus, matrices for which the maximum volume principle cannot guarantee high accuracy are quite rare. A connection of the estimates obtained with the methods for finding the submatrix of the maximum volume and the maximum projective volume is also considered. Numerical experiments show the closeness of theoretical estimates and practical results of fast cross approximation.

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Funding

This work was supported by the Department of the Moscow Center for Fundamental and Applied Mathematics at the Institute of Numerical Mathematics, Russian Academy of Sciences (agreement no. 075-15-2019-1624 with the Ministry of Education and Science of the Russian Federation).

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Authors and Affiliations

  1. Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, 119333, Moscow, Russia

    N. L. Zamarashkin & A. I. Osinsky

  2. Skolkovo Institute of Science and Technology, 121205, Moscow, Russia

    A. I. Osinsky

Authors
  1. N. L. Zamarashkin
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  2. A. I. Osinsky
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Correspondence to N. L. Zamarashkin or A. I. Osinsky.

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Translated by E. Chernokozhin

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Zamarashkin, N.L., Osinsky, A.I. On the Accuracy of Cross and Column Low-Rank Maxvol Approximations in Average. Comput. Math. and Math. Phys. 61, 786–798 (2021). https://doi.org/10.1134/S0965542521050171

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  • DOI: https://doi.org/10.1134/S0965542521050171

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