Faster Kernel Matrix Algebra via Density Estimation

We study fast algorithms for computing fundamental properties of a positive semidefinite kernel matrix K ∈ ℝⁿ×ⁿ corresponding to n points x₁, ..., x_n ∈ 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 + ε relative error in time sublinear in n and linear in d for many popular kernels, including the Gaussian, exponential, and rational quadratic. For these kernels, we also show that the top eigenvalue (and an approximate eigenvector) can be approximated to 1 + ε relative error in time subquadratic in n and linear in d. Our results 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.