  Statistical leverage scores emerged as a fundamental tool for matrix
sketching and column sampling with applications to low rank approximation,
regression, random feature learning and quadrature. Yet, the very nature of
this quantity is barely understood. Borrowing ideas from the orthogonal
polynomial literature, we introduce the regularized Christoffel function
associated to a positive definite kernel. This uncovers a variational
formulation for leverage scores for kernel methods and allows to elucidate
their relationships with the chosen kernel as well as population density. Our
main result quantitatively describes a decreasing relation between leverage
score and population density for a broad class of kernels on Euclidean spaces.
Numerical simulations support our findings.
