An improved estimate for the condition number anomaly of univariate Gaussian correlation matrices

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Abstract

In this short note, it is proved that the derivatives of the parametrized univariate Gaussian correlation matrix R_g (θ) = (exp(−θ(x_i − x_j )^2_{i,j} ∈ R^{n×n} are rank-deficient in the limit θ = 0 up to any order m < (n − 1)/2. This result generalizes the rank deficiency theorem for Euclidean distance matrices, which appear as the first-order derivatives of the Gaussian correlation matrices in the limit θ = 0. As a consequence, it is shown that the condition number of R_g(θ) grows at least as fast as 1(/θ^(mˆ +1) for θ → 0, where mˆ is the largest integer such that mˆ < (n − 1)/2. This considerably improves the previously known growth rate estimate of 1/θ^22 for the so-called Gaussian condition number anomaly.
Original languageEnglish
JournalThe Electronic Journal of Linear Algebra
Volume30
Pages (from-to)592-598
ISSN1081-3810
DOIs
Publication statusPublished - 2015
Externally publishedYes

Keywords

  • Euclidean Distance Matrix
  • Gaussian Correlation Matrix
  • almost negative definite matrix
  • condition number
  • Radial basis functions
  • Condition number
  • Gaussian correlation matrix
  • Euclidean distance matrix
  • Almost negative definite matrix
  • Kriging
  • Vandermonde matrix

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