{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article16777","name":"Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation","abstract":"For any real numbers B \u2265 1 and \u03b4 \u2208 (0,1) and function f: [0,B] \u2192 \u211d, let d_{B; \u03b4}(f) \u2208 \u2124_{> 0} denote the minimum degree of a polynomial p(x) satisfying sup_{x \u2208 [0,B]} |p(x) - f(x)| < \u03b4. In this paper, we provide precise asymptotics for d_{B; \u03b4}(e^{-x}) and d_{B; \u03b4}(e^x) in terms of both B and \u03b4, improving both the previously known upper bounds and lower bounds. In particular, we show d_{B; \u03b4}(e^{-x}) = \u0398(max{\u221a{B log(\u03b4^{-1})}, log(\u03b4^{-1})\/{log(B^{-1} log(\u03b4^{-1}))}}), and d_{B; \u03b4}(e^{x}) = \u0398(max{B, log(\u03b4^{-1})\/{log(B^{-1} log(\u03b4^{-1}))}}), and we explicitly determine the leading coefficients in most parameter regimes.\r\nPolynomial approximations for e^{-x} and e^x have applications to the design of algorithms for many problems, including in scientific computing, graph algorithms, machine learning, and statistics. Our degree bounds show both the power and limitations of these algorithms.\r\nWe focus in particular on the Batch Gaussian Kernel Density Estimation problem for n sample points in \u0398(log n) dimensions with error \u03b4 = n^{-\u0398(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 = \u0398(log n) mirroring the corresponding transition in d_{B; \u03b4}(e^{-x}): \r\n- When B = o(log n), we give the first algorithm running in time n^{1 + o(1)}. \r\n- When B = \u03ba log n for a small constant \u03ba > 0, we give an algorithm running in time n^{1 + O(log log \u03ba^{-1} \/log \u03ba^{-1})}. The log log \u03ba^{-1} \/log \u03ba^{-1} term in the exponent comes from analyzing the behavior of the leading constant in our computation of d_{B; \u03b4}(e^{-x}). \r\n- When B = \u03c9(log n), we show that time n^{2 - o(1)} is necessary assuming SETH.","keywords":["polynomial approximation","kernel density estimation","Chebyshev polynomials"],"author":[{"@type":"Person","name":"Aggarwal, Amol","givenName":"Amol","familyName":"Aggarwal","email":"mailto:amolaggarwal@math.columbia.edu","affiliation":"Department of Mathematics, Columbia University, New York, NY, USA","funding":"Partially supported by NSF grants DGE-1144152 and DMS-1664619, a Harvard Merit\/Graduate Society Term-time Research Fellowship, and a Clay Research Fellowship."},{"@type":"Person","name":"Alman, Josh","givenName":"Josh","familyName":"Alman","email":"mailto:josh@cs.columbia.edu","affiliation":"Department of Computer Science, Columbia University, New York, NY, USA","funding":"Partially supported by a Harvard Michael O. Rabin postdoctoral fellowship and a grant from the Simons Foundation (Grant Number 825870 JA)."}],"position":22,"pageStart":"22:1","pageEnd":"22:23","dateCreated":"2022-07-11","datePublished":"2022-07-11","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Aggarwal, Amol","givenName":"Amol","familyName":"Aggarwal","email":"mailto:amolaggarwal@math.columbia.edu","affiliation":"Department of Mathematics, Columbia University, New York, NY, USA","funding":"Partially supported by NSF grants DGE-1144152 and DMS-1664619, a Harvard Merit\/Graduate Society Term-time Research Fellowship, and a Clay Research Fellowship."},{"@type":"Person","name":"Alman, Josh","givenName":"Josh","familyName":"Alman","email":"mailto:josh@cs.columbia.edu","affiliation":"Department of Computer Science, Columbia University, New York, NY, USA","funding":"Partially supported by a Harvard Michael O. Rabin postdoctoral fellowship and a grant from the Simons Foundation (Grant Number 825870 JA)."}],"copyrightYear":"2022","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.CCC.2022.22","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["https:\/\/doi.org\/10.1109\/FOCS46700.2020.00057","http:\/\/arxiv.org\/abs\/1512.04906","http:\/\/arxiv.org\/abs\/1912.07673"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6437","volumeNumber":234,"name":"37th Computational Complexity Conference (CCC 2022)","dateCreated":"2022-07-11","datePublished":"2022-07-11","editor":{"@type":"Person","name":"Lovett, Shachar","givenName":"Shachar","familyName":"Lovett","email":"mailto:slovett@cs.ucsd.edu","sameAs":"https:\/\/orcid.org\/0000-0003-4552-1443","affiliation":"University of California San Diego, La Jolla, CA, US"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article16777","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6437"}}}