The spectrum of kernel random matrices

Understand

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.