Faster Kernel Matrix Algebra via Density Estimation

• URL: http://arxiv.org/abs/2102.08341v1
• Date: Tue, 16 Feb 2021 18:25:47 GMT
• Title: Faster Kernel Matrix Algebra via Density Estimation
• Authors: Arturs Backurs and Piotr Indyk and Cameron Musco and Tal Wagner
• Abstract summary: We study fast algorithms for computing fundamental properties of a positive semidefinite kernel matrix $K in mathbbRn times n$ corresponding to $n$ points.
• Score: 46.253698241653254
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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