New Nearly-Optimal Coreset for Kernel Density Estimation

07/15/2020

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by   Wai Ming Tai, et al.

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Given a point set P⊂ℝ^d, kernel density estimation for Gaussian kernel is defined as 𝒢_P(x) = 1/|P|∑_p∈ Pe^-‖ x-p ‖^2 for any x∈ℝ^d. We study how to construct a small subset Q of P such that the kernel density estimation of P can be approximated by the kernel density estimation of Q. This subset Q is called coreset. The primary technique in this work is to construct ± 1 coloring on the point set P by the discrepancy theory and apply this coloring algorithm recursively. Our result leverages Banaszczyk's Theorem. When d>1 is constant, our construction gives a coreset of size O(1/ε√(loglog1/ε)) as opposed to the best-known result of O(1/ε√(log1/ε)). It is the first to give a breakthrough on the barrier of √(log) factor even when d=2.