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We construct near-optimal coresets for kernel density estimates for points in $mathbb{R}^d$ when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size $O(sqrt{d}/varepsiloncdot sqrt{log 1/varepsilon} )$, and we show a near-matching lower bound of size $Omega(min{sqrt{d}/varepsilon, 1/varepsilon^2})$. When $dgeq 1/varepsilon^2$, it is known that the size of coreset can be $O(1/varepsilon^2)$. The upper bound is a polynomial-in-$(1/varepsilon)$ improvement when $d in [3,1/varepsilon^2)$ 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.
We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For characteristic k
Given a point set $Psubset mathbb{R}^d$, a kernel density estimation for Gaussian kernel is defined as $overline{mathcal{G}}_P(x) = frac{1}{left|Pright|}sum_{pin P}e^{-leftlVert x-p rightrVert^2}$ for any $xinmathbb{R}^d$. We study how to construct a
Many clustering algorithms are guided by certain cost functions such as the widely-used $k$-means cost. These algorithms divide data points into clusters with often complicated boundaries, creating difficulties in explaining the clustering decision.
Coreset is usually a small weighted subset of $n$ input points in $mathbb{R}^d$, that provably approximates their loss function for a given set of queries (models, classifiers, etc.). Coresets become increasingly common in machine learning since exis