Given a set of n points in d dimensions, the Euclidean k -means problem (resp. Euclidean k -median) consists of finding k centers such that the sum of squared distances (resp. The arguably most popular way of dealing with this problem in the big data setting is to first compress the data by computing a weighted subset known as a coreset and then run any algorithm on this subset. The guarantee of the coreset is that for any candidate solution, the ratio between coreset cost and the cost of the original instance is less than a (1\pm \varepsilon) factor. The best known lower bound for both problems is \Omega(k\varepsilon {-2}) .

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 kernels (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 $d \geq 3$.