Leveraging Well-Conditioned Bases: Streaming and Distributed Summaries in Minkowski $p$-Norms

Charlie Dickens, Graham Cormode, David Woodruff
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:1243-1251, 2018.

Abstract

Work on approximate linear algebra has led to efficient distributed and streaming algorithms for problems such as approximate matrix multiplication, low rank approximation, and regression, primarily for the Euclidean norm $\ell_2$. We study other $\ell_p$ norms, which are more robust for $p < 2$, and can be used to find outliers for $p > 2$. Unlike previous algorithms for such norms, we give algorithms that are (1) deterministic, (2) work simultaneously for every $p \geq 1$, including $p = \infty$, and (3) can be implemented in both distributed and streaming environments. We study $\ell_p$-regression, entrywise $\ell_p$-low rank approximation, and versions of approximate matrix multiplication.

Cite this Paper

BibTeX
@InProceedings{pmlr-v80-dickens18a, title = {Leveraging Well-Conditioned Bases: Streaming and Distributed Summaries in {M}inkowski $p$-Norms}, author = {Dickens, Charlie and Cormode, Graham and Woodruff, David}, booktitle = {Proceedings of the 35th International Conference on Machine Learning}, pages = {1243--1251}, year = {2018}, editor = {Dy, Jennifer and Krause, Andreas}, volume = {80}, series = {Proceedings of Machine Learning Research}, month = {10--15 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v80/dickens18a/dickens18a.pdf}, url = {https://proceedings.mlr.press/v80/dickens18a.html}, abstract = {Work on approximate linear algebra has led to efficient distributed and streaming algorithms for problems such as approximate matrix multiplication, low rank approximation, and regression, primarily for the Euclidean norm $\ell_2$. We study other $\ell_p$ norms, which are more robust for $p < 2$, and can be used to find outliers for $p > 2$. Unlike previous algorithms for such norms, we give algorithms that are (1) deterministic, (2) work simultaneously for every $p \geq 1$, including $p = \infty$, and (3) can be implemented in both distributed and streaming environments. We study $\ell_p$-regression, entrywise $\ell_p$-low rank approximation, and versions of approximate matrix multiplication.} }
Endnote
%0 Conference Paper %T Leveraging Well-Conditioned Bases: Streaming and Distributed Summaries in Minkowski $p$-Norms %A Charlie Dickens %A Graham Cormode %A David Woodruff %B Proceedings of the 35th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2018 %E Jennifer Dy %E Andreas Krause %F pmlr-v80-dickens18a %I PMLR %P 1243--1251 %U https://proceedings.mlr.press/v80/dickens18a.html %V 80 %X Work on approximate linear algebra has led to efficient distributed and streaming algorithms for problems such as approximate matrix multiplication, low rank approximation, and regression, primarily for the Euclidean norm $\ell_2$. We study other $\ell_p$ norms, which are more robust for $p < 2$, and can be used to find outliers for $p > 2$. Unlike previous algorithms for such norms, we give algorithms that are (1) deterministic, (2) work simultaneously for every $p \geq 1$, including $p = \infty$, and (3) can be implemented in both distributed and streaming environments. We study $\ell_p$-regression, entrywise $\ell_p$-low rank approximation, and versions of approximate matrix multiplication.
APA
Dickens, C., Cormode, G. & Woodruff, D.. (2018). Leveraging Well-Conditioned Bases: Streaming and Distributed Summaries in Minkowski $p$-Norms. Proceedings of the 35th International Conference on Machine Learning, in Proceedings of Machine Learning Research 80:1243-1251 Available from https://proceedings.mlr.press/v80/dickens18a.html.

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