Near-Optimal Coresets of Kernel Density Estimates - Phillips, Jeff M.; Tai, Wai Ming (Discrete & computational geometry, 2020)
We construct near-optimal coresets for kernel density estimates for points in R d when the kernel is positive definite. Specifically we provide a polynomial time construction for a coreset of size O ( d / ε · log 1 / ε ) , and we show a near-matching lower bound of size Ω ( min { d / ε , 1 / ε 2 } ) . When d ≥ 1 / ε 2 , it is known that the size of coreset can be O ( 1 / ε 2 ) . The upper bound is a polynomial-in- ( 1 / ε ) improvement when d ∈ [ 3 , 1 / ε 2 ) 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.