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Description:
Given a point set P ⊂ ℝ^d, the kernel density estimate of P is defined as 𝒢-_P(x) = 1/|P| ∑_{p ∈ P}e^{-∥x-p∥²} for any x ∈ ℝ^d. We study how to construct a small subset Q of P such that the kernel density estimate of P is approximated by the kernel density estimate of Q. This subset Q is called a coreset. The main technique in this work is constructing a ± 1 coloring on the point set P by discrepancy theory and we leverage Banaszczyk’s Theorem. When d > 1 is a constant, our construction gives a coreset of size O(1/ε) as opposed to the best-known result of O(1/ε √{log 1/ε}). It is the first result to give a breakthrough on the barrier of √log factor even when d = 2.