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Keywords

Euclidean distance matrix, Gaussian correlation matrix, almost negative definite matrix, spatial linear model, radial basis functions, Vandermonde matrix, condition number

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.

abs_vol30_pp592-598.pdf (31 kB)
Abstract

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