We relate the problem of best low-rank approximation in the spectral norm for a matrix A to Kolmogorov n-widths and corresponding optimal spaces. We characterize all the optimal spaces for the image of the Euclidean unit ball under A, and we show that any orthonormal basis in an n-dimensional optimal space generates a best rank-n approximation to A. We also present a simple and explicit construction to obtain a sequence of optimal n-dimensional spaces once an initial optimal space is known. This results in a variety of solutions to the best low-rank approximation problem and provides alternatives to the truncated singular value decomposition. This variety can be exploited to obtain best low-rank approximations with problem-oriented properties.

Floater, M.s., Manni, C., Sande, E., Speleers, H. (2021). Best low-rank approximations and Kolmogorov n-widths. SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 42(1), 330-350 [10.1137/20M1355720].

Best low-rank approximations and Kolmogorov n-widths

Manni C.;Speleers H.
2021-03-01

Abstract

We relate the problem of best low-rank approximation in the spectral norm for a matrix A to Kolmogorov n-widths and corresponding optimal spaces. We characterize all the optimal spaces for the image of the Euclidean unit ball under A, and we show that any orthonormal basis in an n-dimensional optimal space generates a best rank-n approximation to A. We also present a simple and explicit construction to obtain a sequence of optimal n-dimensional spaces once an initial optimal space is known. This results in a variety of solutions to the best low-rank approximation problem and provides alternatives to the truncated singular value decomposition. This variety can be exploited to obtain best low-rank approximations with problem-oriented properties.
mar-2021
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/08 - ANALISI NUMERICA
English
Con Impact Factor ISI
Low-rank approximation; Best approximation; n-Widths; Optimal spaces
Floater, M.s., Manni, C., Sande, E., Speleers, H. (2021). Best low-rank approximations and Kolmogorov n-widths. SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 42(1), 330-350 [10.1137/20M1355720].
Floater, Ms; Manni, C; Sande, E; Speleers, H
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/274802
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