euMMD: efficiently computing the MMD two-sample test statistic for univariate data

摘要

Abstract The maximum mean discrepancy (MMD) test is a nonparametric kernelised two-sample test that, when using a characteristic kernel, can detect any distributional change between two samples. However, when the total number of ddocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$d$$end{document}-dimensional observations is ndocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$n$$end{document}, direct computation of the test statistic is O(dn2)documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathcal {O}(dn^2 )$$end{document}. While approximations with lower computational complexity are known, more efficient methods for computing the exact test statistic are unknown. This paper provides an exact method for computing the MMD test statistic for the univariate case in O(nlogn)documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathcal {O}(nlog n)$$end{document} using the Laplacian kernel. Furthermore, this exact method is extended to an approximate method for ddocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$d$$end{document}-dimensional real-valued data also with complexity log-linear in the number of observations. Experiments show that this approximate method can have good statistical performance when compared to the exact test, particularly in cases where d>ndocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$d> n$$end{document}.