This study investigates the applicability and efficiency of support vector machines for the problem of estimating the earthquake response spectra from the Fourier amplitude spectra of the ground motion acceleration. Two methods are commonly used for this purpose: time domain simulations, and the random vibration theory. The use of time domain simulations offers high accuracy at high computational cost, while the use random vibration theory, although not computationally intensive, requires knowledge of the statistical distribution of the response amplitudes. This study treats the task of estimating response spectra from the Fourier spectra as a nonlinear regression problem, and constructs a supervised machine learning algorithm with minimal sensitivity to noise and outliers. In this method, pairs of vectors consisting of Fourier amplitude spectra and pseudo-velocity response spectra are transformed into a high dimensional feature space where the nonlinear relationship between them can be represented as a line. No assumptions regarding the probability density function of response amplitudes are required. A practical application is presented using artificially generated accelerograms, and it is shown that the support vector machines can predict the response spectra for a wide range of vibration periods.