Data adaptive RKHS Tikhonov regularization for learning kernels in operators

Fei Lu, Quanjun Lang, Qingci An
Proceedings of Mathematical and Scientific Machine Learning, PMLR 190:158-172, 2022.

Abstract

We present DARTR: a Data Adaptive RKHS Tikhonov Regularization method for the linear inverse problem of nonparametric learning of function parameters in operators. A key ingredient is a system intrinsic data adaptive (SIDA) RKHS, whose norm restricts the learning to take place in the function space of identifiability. DARTR utilizes this norm and selects the regularization parameter by the L-curve method. We illustrate its performance in examples including integral operators, nonlinear operators and nonlocal operators with discrete synthetic data. Numerical results show that DARTR leads to an accurate estimator robust to both numerical error due to discrete data and noise in data, and the estimator converges at a consistent rate as the data mesh refines under different levels of noises, outperforming two baseline regularizers using $l^2$ and $L^2$ norms.

Cite this Paper

BibTeX
@InProceedings{pmlr-v190-lu22a, title = {Data adaptive RKHS Tikhonov regularization for learning kernels in operators}, author = {Lu, Fei and Lang, Quanjun and An, Qingci}, booktitle = {Proceedings of Mathematical and Scientific Machine Learning}, pages = {158--172}, year = {2022}, editor = {Dong, Bin and Li, Qianxiao and Wang, Lei and Xu, Zhi-Qin John}, volume = {190}, series = {Proceedings of Machine Learning Research}, month = {15--17 Aug}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v190/lu22a/lu22a.pdf}, url = {https://proceedings.mlr.press/v190/lu22a.html}, abstract = {We present DARTR: a Data Adaptive RKHS Tikhonov Regularization method for the linear inverse problem of nonparametric learning of function parameters in operators. A key ingredient is a system intrinsic data adaptive (SIDA) RKHS, whose norm restricts the learning to take place in the function space of identifiability. DARTR utilizes this norm and selects the regularization parameter by the L-curve method. We illustrate its performance in examples including integral operators, nonlinear operators and nonlocal operators with discrete synthetic data. Numerical results show that DARTR leads to an accurate estimator robust to both numerical error due to discrete data and noise in data, and the estimator converges at a consistent rate as the data mesh refines under different levels of noises, outperforming two baseline regularizers using $l^2$ and $L^2$ norms.} }
Endnote
%0 Conference Paper %T Data adaptive RKHS Tikhonov regularization for learning kernels in operators %A Fei Lu %A Quanjun Lang %A Qingci An %B Proceedings of Mathematical and Scientific Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Bin Dong %E Qianxiao Li %E Lei Wang %E Zhi-Qin John Xu %F pmlr-v190-lu22a %I PMLR %P 158--172 %U https://proceedings.mlr.press/v190/lu22a.html %V 190 %X We present DARTR: a Data Adaptive RKHS Tikhonov Regularization method for the linear inverse problem of nonparametric learning of function parameters in operators. A key ingredient is a system intrinsic data adaptive (SIDA) RKHS, whose norm restricts the learning to take place in the function space of identifiability. DARTR utilizes this norm and selects the regularization parameter by the L-curve method. We illustrate its performance in examples including integral operators, nonlinear operators and nonlocal operators with discrete synthetic data. Numerical results show that DARTR leads to an accurate estimator robust to both numerical error due to discrete data and noise in data, and the estimator converges at a consistent rate as the data mesh refines under different levels of noises, outperforming two baseline regularizers using $l^2$ and $L^2$ norms.
APA
Lu, F., Lang, Q. & An, Q.. (2022). Data adaptive RKHS Tikhonov regularization for learning kernels in operators. Proceedings of Mathematical and Scientific Machine Learning, in Proceedings of Machine Learning Research 190:158-172 Available from https://proceedings.mlr.press/v190/lu22a.html.

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