Sampling-based Nyström Approximation and Kernel Quadrature

Satoshi Hayakawa, Harald Oberhauser, Terry Lyons
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:12678-12699, 2023.

Abstract

We analyze the Nyström approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nyström approximation with i.i.d. sampling and singular-value decomposition in the continuous regime; the proof techniques are borrowed from statistical learning theory. We further introduce a refined selection of subspaces in Nyström approximation with theoretical guarantees that is applicable to non-i.i.d. landmark points. Finally, we discuss their application to convex kernel quadrature and give novel theoretical guarantees as well as numerical observations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-hayakawa23a, title = {Sampling-based Nyström Approximation and Kernel Quadrature}, author = {Hayakawa, Satoshi and Oberhauser, Harald and Lyons, Terry}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {12678--12699}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/hayakawa23a/hayakawa23a.pdf}, url = {https://proceedings.mlr.press/v202/hayakawa23a.html}, abstract = {We analyze the Nyström approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nyström approximation with i.i.d. sampling and singular-value decomposition in the continuous regime; the proof techniques are borrowed from statistical learning theory. We further introduce a refined selection of subspaces in Nyström approximation with theoretical guarantees that is applicable to non-i.i.d. landmark points. Finally, we discuss their application to convex kernel quadrature and give novel theoretical guarantees as well as numerical observations.} }
Endnote
%0 Conference Paper %T Sampling-based Nyström Approximation and Kernel Quadrature %A Satoshi Hayakawa %A Harald Oberhauser %A Terry Lyons %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-hayakawa23a %I PMLR %P 12678--12699 %U https://proceedings.mlr.press/v202/hayakawa23a.html %V 202 %X We analyze the Nyström approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nyström approximation with i.i.d. sampling and singular-value decomposition in the continuous regime; the proof techniques are borrowed from statistical learning theory. We further introduce a refined selection of subspaces in Nyström approximation with theoretical guarantees that is applicable to non-i.i.d. landmark points. Finally, we discuss their application to convex kernel quadrature and give novel theoretical guarantees as well as numerical observations.
APA
Hayakawa, S., Oberhauser, H. & Lyons, T.. (2023). Sampling-based Nyström Approximation and Kernel Quadrature. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:12678-12699 Available from https://proceedings.mlr.press/v202/hayakawa23a.html.

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