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Near-Optimal Coresets of Kernel Density Estimates

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 نشر من قبل Wai Ming Tai
 تاريخ النشر 2018
  مجال البحث الهندسة المعلوماتية
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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.



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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 ernels (including Gaussian and Laplace kernels), our approximation preserves the $L_infty$ error between kernel density estimates within error $epsilon$, with coreset size $2/epsilon^2$, but no other aspects of the data, including the dimension, the diameter of the point set, or the bandwidth of the kernel common to other approximations. When the dimension is unrestricted, we show this bound is tight for these kernels as well as a much broader set. This work provides a careful analysis of the iterative Frank-Wolfe algorithm adapted to this context, an algorithm called emph{kernel herding}. This analysis unites a broad line of work that spans statistics, machine learning, and geometry. When the dimension $d$ is constant, we demonstrate much tighter bounds on the size of the coreset specifically for Gaussian kernels, showing that it is bounded by the size of the coreset for axis-aligned rectangles. Currently the best known constructive bound is $O(frac{1}{epsilon} log^d frac{1}{epsilon})$, and non-constructively, this can be improved by $sqrt{log frac{1}{epsilon}}$. This improves the best constant dimension bounds polynomially for $d geq 3$.
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