Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation

For any real numbers B ≥ 1 and δ ∈ (0, 1) and function f: [0, B] rightarrow ℝ, let dB; δ (f) ∈ mathbbZ> 0 denote the minimum degree of a polynomial p(x) satisfying supx ∈ [0, B] big| p(x) - f(x) big| < δ. In this paper, we provide precise asymptotics for dB; δ (e-x) and dB; δ (ex) in terms of both B and δ, improving both the previously known upper bounds and lower bounds. In particular, we show dB; δ (e-x) = Thetaleft( max left √B log(δ-1), fraclog(δ-1) log(B-1 log(δ-1)) rightright), and dB; δ (ex) = Thetaleft( max left B, fraclog(δ-1) log(B-1 log(δ-1)) rightright). 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 Theta(log n) dimensions with error δ = n-Theta(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 = Theta(log n) mirroring the corresponding transition in dB; δ (e-x): - When B=o(log n), we give the first algorithm running in time n1 + o(1). - When B = kappa log n for a small constant kappa>0, we give an algorithm running in time n^1 + O(log log kappa-1 /log kappa-1). The log log kappa-1 /log kappa-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(log n), we show that time n2 - o(1) is necessary assuming SETH.