DeepAI
Log In Sign Up

The spectral properties of Vandermonde matrices with clustered nodes

09/04/2019
by   Dmitry Batenkov, et al.
0

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 h ≲1 N, and the clusters being separated from each other by at least θ≳1 N. 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 π/2-c_1/N θ-c_2 N h, for some constants c_1,c_2 depending only on the multiplicities of the clusters. As a result, spectral analysis of V_N 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

09/03/2018

Stability of partial Fourier matrices with clustered nodes

We prove sharp lower bounds for the smallest singular value of a partial...
07/16/2019

On the smallest singular value of multivariate Vandermonde matrices with clustered nodes

We prove lower bounds for the smallest singular value of rectangular, mu...
08/22/2018

Clustered Planarity = Flat Clustered Planarity

The complexity of deciding whether a clustered graph admits a clustered ...
09/11/2021

A Note on Projection-Based Recovery of Clusters in Markov Chains

Let T_0 be the transition matrix of a purely clustered Markov chain, i.e...
03/15/2021

Multivariate Vandermonde matrices with separated nodes on the unit circle are stable

We prove explicit lower bounds for the smallest singular value and upper...
10/04/2019

C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

For a clustered graph, i.e, a graph whose vertex set is recursively part...