Asymptotic Spectral Behavior of Kernel Matrices in Complex Valued Observations

The spectral behavior of kernel matrices built from complex multi-variate data is established in the asymptotic regime where both the number of observations and their dimensionality increase without bound at the same rate. The result is an extension of currently available results for inner product based kernel matrices formed from real valued observations to the case where the input data is complex valued. In particular, assuming complex independent standardized Gaussian inputs and imposing certain conditions on the kernel function, it is shown that the empirical distribution of eigenvalues of this type of matrices converges almost surely to a probability measure in this asymptotic domain. Furthermore, the asymptotic spectral density can be obtained by solving a quartic polynomial equation involving its Stieltjes transform and some coefficients depending on the Hermite-like expansion of the kernel function. This is in stark contrast with the equivalent result for real valued observations, in which the underlying polynomial equation is cubic.