If I'm not mistaken, Gaussian Processes are expensive for large n_samples,
not for large n_features. The reason is because the kernel matrix (called
covariance matrix in the GP literature) needs to be inversed, which takes
O(n_samples^3) complexity with a Cholesky decomposition. That said, kernels
methods like SVMs or Gaussian Processes are usually not used much with
high-dimensional data. Kernels are useful to implicitly project
low-dimensional data to higher (even infinite) dimensional spaces. If your
data is already high-dimensional, there's nothing to gain from using
kernels. A good example is text classification, where everyone is using
linear kernels.
HTH,
Mathieu
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