  We study fast algorithms for computing fundamental properties of a positive
semidefinite kernel matrix $K \in \mathbb{R}^{n \times n}$ corresponding to $n$
points $x_1,\ldots,x_n \in \mathbb{R}^d$. In particular, we consider estimating
the sum of kernel matrix entries, along with its top eigenvalue and
eigenvector.
  We show that the sum of matrix entries can be estimated to $1+\epsilon$
relative error in time $sublinear$ in $n$ and linear in $d$ for many popular
kernels, including the Gaussian, exponential, and rational quadratic kernels.
For these kernels, we also show that the top eigenvalue (and an approximate
eigenvector) can be approximated to $1+\epsilon$ relative error in time
$subquadratic$ in $n$ and linear in $d$.
  Our algorithms represent significant advances in the best known runtimes for
these problems. They leverage the positive definiteness of the kernel matrix,
along with a recent line of work on efficient kernel density estimation.
