Integration in reproducing kernel Hilbert spaces of Gaussian kernels
نویسندگان
چکیده
The Gaussian kernel plays a central role in machine learning, uncertainty quantification and scattered data approximation, but has received relatively little attention from numerical analysis standpoint. basic problem of finding an algorithm for efficient integration functions reproduced by kernels not been fully solved. In this article we construct two classes algorithms that use N N evaluations to integrate alttext="d"> d encoding="application/x-tex">d -variate prove the exponential or super-algebraic decay their worst-case errors. contrast earlier work, no constraints are placed on length-scale parameter kernel. first class is obtained via appropriate scaling classical Gauss–Hermite rules. For these derive lower upper bounds error forms alttext="exp left-parenthesis minus c 1 N Superscript slash d Baseline right-parenthesis 4 right-parenthesis"> exp ( −<!-- − <mml:msub> c 1 / stretchy="false">) 4 encoding="application/x-tex">\exp (-c_1 N^{1/d}) N^{1/(4d)} 2 negative 2 (-c_2 N^{-1/(4d)} , respectively, positive constants alttext="c greater-than 2"> > encoding="application/x-tex">c_1 > c_2 . second more flexible uses optimal weights points may be taken as nested sequence. form 3 3 (-c_3 N^{1/(2d)}) constant 3"> encoding="application/x-tex">c_3
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2021
ISSN: ['1088-6842', '0025-5718']
DOI: https://doi.org/10.1090/mcom/3659