Faster Kernel Matrix Algebra via Density Estimation

Abstract

We study fast algorithms for computing fundamental properties of a positive semidefinite kernel matrix K R^n n corresponding to n points x_1,,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+ 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+ 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.

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Reproductions