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

Abstract

In this short note, it is proved that the derivatives of the parametrized univariate Gaussian correlation matrix R_g (θ) = (exp(âθ(x_i â x_j )^2_{i,j} â R^{nÃn} are rank-deficient in the limit θ = 0 up to any order m < (n â 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^(mË +1) for θ â 0, where mË is the largest integer such that mË < (n â 1)/2. This considerably improves the previously known growth rate estimate of 1/θ^22 for the so-called Gaussian condition number anomaly.