Efficient computation of smoothing splines via adaptive basis sampling

Type
Article

Authors
Ma, Ping
Huang, Jianhua Z.
Zhang, Nan

Online Publication Date
2015-06-24

Print Publication Date
2015-09

Date
2015-06-24

Abstract
© 2015 Biometrika Trust. Smoothing splines provide flexible nonparametric regression estimators. However, the high computational cost of smoothing splines for large datasets has hindered their wide application. In this article, we develop a new method, named adaptive basis sampling, for efficient computation of smoothing splines in super-large samples. Except for the univariate case where the Reinsch algorithm is applicable, a smoothing spline for a regression problem with sample size n can be expressed as a linear combination of n basis functions and its computational complexity is generally O(n). We achieve a more scalable computation in the multivariate case by evaluating the smoothing spline using a smaller set of basis functions, obtained by an adaptive sampling scheme that uses values of the response variable. Our asymptotic analysis shows that smoothing splines computed via adaptive basis sampling converge to the true function at the same rate as full basis smoothing splines. Using simulation studies and a large-scale deep earth core-mantle boundary imaging study, we show that the proposed method outperforms a sampling method that does not use the values of response variables.

Citation
Ma P, Huang JZ, Zhang N (2015) Efficient computation of smoothing splines via adaptive basis sampling. Biometrika 102: 631–645. Available: http://dx.doi.org/10.1093/biomet/asv009.

Acknowledgements
The first author thanks Chong Gu for many helpful discussions. Ma’s work was partially supportedby the National Science Foundation and the U.S. Department of Energy. Huang’s workwas partially supported by the National Science Foundation and King Abdullah University ofScience and Technology.

Publisher
Oxford University Press (OUP)

Journal
Biometrika

DOI
10.1093/biomet/asv009

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