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I want to know what are the specific areas in which Fourier methods are used in machine learning. Apart from feature extraction and spectral analysis, I want to know if there are any learning algorithms that are based on Fourier methods.

I also want to know if there is any motivation for using Fourier methods for probabilistic graphical models.

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A couple things come to mind...

Performing convolutions efficiently as products in the Fourier domain. An example would be training large convolutional neural nets.

For example, see: Fast Training of Convolutional Networks through FFTs (Mathieu et al. 2013)

Another application is sparse signal processing, where the goal is to approximate a signal as a sparse linear combination of basis functions from a 'signal dictionary'. The link here is that the set of sinusoids are, of course, a good dictionary for signals that are sparse in the Fourier domain. If I recall correctly, Fourier dictionaries show up in this literature.

On a related note, you should also be able to find Fourier methods in the compressed sensing literature

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In a theory of random processes we use Fourier transform to get the spectral density of a covariance function.

Then spectral density can be used to verify that the function is a covariance function (Bochner-Khinchin's theorem). Also spectral density is useful while proving theoretical results about quality of Gaussian process regression models (see recent Van der Vaart's works or Stein's book on interpolation for spatial data).

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