  We place ourselves in the setting of high-dimensional statistical inference
where the number of variables $p$ in a dataset of interest is of the same order
of magnitude as the number of observations $n$. We consider the spectrum of
certain kernel random matrices, in particular $n\times n$ matrices whose
$(i,j)$th entry is $f(X_i'X_j/p)$ or $f(\Vert X_i-X_j\Vert^2/p)$ where $p$ is
the dimension of the data, and $X_i$ are independent data vectors. Here $f$ is
assumed to be a locally smooth function. The study is motivated by questions
arising in statistics and computer science where these matrices are used to
perform, among other things, nonlinear versions of principal component
analysis. Surprisingly, we show that in high-dimensions, and for the models we
analyze, the problem becomes essentially linear--which is at odds with
heuristics sometimes used to justify the usage of these methods. The analysis
also highlights certain peculiarities of models widely studied in random matrix
theory and raises some questions about their relevance as tools to model
high-dimensional data encountered in practice.
