Error Estimation for Sketched SVD via the Bootstrap

Miles Lopes, N. Benjamin Erichson, Michael Mahoney
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:6382-6392, 2020.

Abstract

In order to compute fast approximations to the singular value decompositions (SVD) of very large matrices, randomized sketching algorithms have become a leading approach. However, a key practical difficulty of sketching an SVD is that the user does not know how far the sketched singular vectors/values are from the exact ones. Indeed, the user may be forced to rely on analytical worst-case error bounds, which may not account for the unique structure of a given problem. As a result, the lack of tools for error estimation often leads to much more computation than is really necessary. To overcome these challenges, this paper develops a fully data-driven bootstrap method that numerically estimates the actual error of sketched singular vectors/values. Furthermore, the method is computationally inexpensive, because it operates only on sketched objects, and hence it requires no extra passes over the full matrix being factored.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-lopes20a, title = {Error Estimation for Sketched {SVD} via the Bootstrap}, author = {Lopes, Miles and Erichson, N. Benjamin and Mahoney, Michael}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {6382--6392}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/lopes20a/lopes20a.pdf}, url = {https://proceedings.mlr.press/v119/lopes20a.html}, abstract = {In order to compute fast approximations to the singular value decompositions (SVD) of very large matrices, randomized sketching algorithms have become a leading approach. However, a key practical difficulty of sketching an SVD is that the user does not know how far the sketched singular vectors/values are from the exact ones. Indeed, the user may be forced to rely on analytical worst-case error bounds, which may not account for the unique structure of a given problem. As a result, the lack of tools for error estimation often leads to much more computation than is really necessary. To overcome these challenges, this paper develops a fully data-driven bootstrap method that numerically estimates the actual error of sketched singular vectors/values. Furthermore, the method is computationally inexpensive, because it operates only on sketched objects, and hence it requires no extra passes over the full matrix being factored.} }
Endnote
%0 Conference Paper %T Error Estimation for Sketched SVD via the Bootstrap %A Miles Lopes %A N. Benjamin Erichson %A Michael Mahoney %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-lopes20a %I PMLR %P 6382--6392 %U https://proceedings.mlr.press/v119/lopes20a.html %V 119 %X In order to compute fast approximations to the singular value decompositions (SVD) of very large matrices, randomized sketching algorithms have become a leading approach. However, a key practical difficulty of sketching an SVD is that the user does not know how far the sketched singular vectors/values are from the exact ones. Indeed, the user may be forced to rely on analytical worst-case error bounds, which may not account for the unique structure of a given problem. As a result, the lack of tools for error estimation often leads to much more computation than is really necessary. To overcome these challenges, this paper develops a fully data-driven bootstrap method that numerically estimates the actual error of sketched singular vectors/values. Furthermore, the method is computationally inexpensive, because it operates only on sketched objects, and hence it requires no extra passes over the full matrix being factored.
APA
Lopes, M., Erichson, N.B. & Mahoney, M.. (2020). Error Estimation for Sketched SVD via the Bootstrap. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:6382-6392 Available from https://proceedings.mlr.press/v119/lopes20a.html.

Related Material