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Near-Optimal Coresets of Kernel Density Estimates

Authors Jeff M. Phillips, Wai Ming Tai

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LIPIcs.SoCG.2018.66.pdf
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Jeff M. Phillips
Wai Ming Tai

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Jeff M. Phillips and Wai Ming Tai. Near-Optimal Coresets of Kernel Density Estimates. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 66:1-66:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.SoCG.2018.66

Abstract

We construct near-optimal coresets for kernel density estimate for points in R^d when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size O(sqrt{d log (1/epsilon)}/epsilon), and we show a near-matching lower bound of size Omega(sqrt{d}/epsilon). The upper bound is a polynomial in 1/epsilon improvement when d in [3,1/epsilon^2) (for all kernels except the Gaussian kernel which had a previous upper bound of O((1/epsilon) log^d (1/epsilon))) and the lower bound is the first known lower bound to depend on d for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.

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Keywords
  • Coresets
  • Kernel Density Estimate
  • Discrepancy

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