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Title: Faster Kernel Matrix Algebra via Density Estimation
We study fast algorithms for computing basic properties of an n x n positive semidefinite kernel matrix K corresponding to n points x_1,...,x_n in R^d. In particular, we consider the estimating the sum of kernel matrix entries, along with its top eigenvalue and eigenvector. These are some of the most basic problems defined over kernel matrices. We show that the sum of matrix entries can be estimated up to a multiplicative factor of 1+epsilon in time sublinear in n and linear in d for many popular kernel functions, including the Gaussian, exponential, and rational quadratic kernels. For these kernels, we also show that the top eigenvalue (and a witnessing approximate eigenvector) can be approximated to a multiplicative factor of 1+epsilon in time sub-quadratic 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.  more » « less
Award ID(s):
2022448 2006798 1740751
PAR ID:
10276627
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
International Conference on Machine Learning
Page Range / eLocation ID:
500-510
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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