  This article investigates the eigenspectrum of the inner product-type kernel
matrix $\sqrt{p} \mathbf{K}=\{f( \mathbf{x}_i^{\sf T}
\mathbf{x}_j/\sqrt{p})\}_{i,j=1}^n $ under a binary mixture model in the high
dimensional regime where the number of data $n$ and their dimension $p$ are
both large and comparable. Based on recent advances in random matrix theory, we
show that, for a wide range of nonlinear functions $f$, the eigenspectrum
behavior is asymptotically equivalent to that of an (at most) cubic function.
This sheds new light on the understanding of nonlinearity in large dimensional
problems. As a byproduct, we propose a simple function prototype valued in $
(-1,0,1) $ that, while reducing both storage memory and running time, achieves
the same (asymptotic) classification performance as any arbitrary function $f$.
