  Random feature maps are ubiquitous in modern statistical machine learning,
where they generalize random projections by means of powerful, yet often
difficult to analyze nonlinear operators. In this paper, we leverage the
"concentration" phenomenon induced by random matrix theory to perform a
spectral analysis on the Gram matrix of these random feature maps, here for
Gaussian mixture models of simultaneously large dimension and size. Our results
are instrumental to a deeper understanding on the interplay of the nonlinearity
and the statistics of the data, thereby allowing for a better tuning of random
feature-based techniques.
