Best Low-Rank Approximations And Kolmogorov N-Widths
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS(2021)
摘要
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.
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关键词
low-rank approximation, best approximation, n-widths, optimal spaces
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