TY - JOUR

T1 - The spectral properties of Vandermonde matrices with clustered nodes

AU - Batenkov, Dmitry

AU - Diederichs, Benedikt

AU - Goldman, Gil

AU - Yomdin, Yosef

N1 - Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2021/1/15

Y1 - 2021/1/15

N2 - We study rectangular Vandermonde matrices V with N+1 rows and s irregularly spaced nodes on the unit circle, in cases where some of the nodes are “clustered” together – the elements inside each cluster being separated by at most [Formula presented], and the clusters being separated from each other by at least [Formula presented]. We show that any pair of column subspaces corresponding to two different clusters are nearly orthogonal: the minimal principal angle between them is at most [Formula presented] for some constants c1,c2 depending only on the multiplicities of the clusters. As a result, spectral analysis of VN is significantly simplified by reducing the problem to the analysis of each cluster individually. Consequently we derive accurate estimates for 1) all the singular values of V, and 2) componentwise condition numbers for the linear least squares problem. Importantly, these estimates are exponential only in the local cluster multiplicities, while changing at most linearly with s.

AB - We study rectangular Vandermonde matrices V with N+1 rows and s irregularly spaced nodes on the unit circle, in cases where some of the nodes are “clustered” together – the elements inside each cluster being separated by at most [Formula presented], and the clusters being separated from each other by at least [Formula presented]. We show that any pair of column subspaces corresponding to two different clusters are nearly orthogonal: the minimal principal angle between them is at most [Formula presented] for some constants c1,c2 depending only on the multiplicities of the clusters. As a result, spectral analysis of VN is significantly simplified by reducing the problem to the analysis of each cluster individually. Consequently we derive accurate estimates for 1) all the singular values of V, and 2) componentwise condition numbers for the linear least squares problem. Importantly, these estimates are exponential only in the local cluster multiplicities, while changing at most linearly with s.

KW - Condition number

KW - Nonuniform Fourier matrices

KW - Singular values

KW - Sub-Rayleigh resolution

KW - Subspace angles

KW - Super-resolution

KW - Vandermonde matrices with nodes on the unit circle

UR - http://www.scopus.com/inward/record.url?scp=85090354117&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2020.08.034

DO - 10.1016/j.laa.2020.08.034

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AN - SCOPUS:85090354117

VL - 609

SP - 37

EP - 72

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -