Low-Rank Matrix Approximations Do Not Need a Singular Value Gap

2018/01/02 by Drineas, Petros, Ipsen, Ilse C. F. · 1 citation
#15A12 #15A18 #15A42 #65F15 #65F35 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1801.00670

Abstract

This is a systematic investigation into the sensitivity of low-rank approximations of real matrices. We show that the low-rank approximation errors, in the two-norm, Frobenius norm and more generally, any Schatten p-norm, are insensitive to additive rank-preserving perturbations in the projector basis; and to matrix perturbations that are additive or change the number of columns (including multiplicative perturbations). Thus, low-rank matrix approximations are always well-posed and do not require a singular value gap. In the presence of a singular value gap, connections are established between low-rank approximations and subspace angles.

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