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Creators/Authors contains: "Aggarwal, Amol and"

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  1. (, Computational complexity)
    Polynomial approximations for e−x and ex have applications to the design of algorithms for many problems, and our degree bounds show both the power and limitations of these algorithms. We focus in particular on the Batch Gaussian Kernel Density Estimation problem for n sample points in Θ(logn) dimensions with error δ=n−Θ(1). We show that the running time one can achieve depends on the square of the diameter of the point set, B, with a transition at B=Θ(logn) mirroring the corresponding transition in dB;δ(e−x): - When B=o(logn), we give the first algorithm running in time n1+o(1). - When B=κlogn for a small constant κ>0, we give an algorithm running in time n1+O(loglogκ−1/logκ−1). The loglogκ−1/logκ−1 term in the exponent comes from analyzing the behavior of the leading constant in our computation of dB;δ(e−x). - When B=ω(logn), we show that time n2−o(1) is necessary assuming SETH. 
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