Nearly-Linear Time and Streaming Algorithms for Outlier-Robust PCA

Ilias Diakonikolas, Daniel Kane, Ankit Pensia, Thanasis Pittas
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:7886-7921, 2023.

Abstract

We study principal component analysis (PCA), where given a dataset in $\mathbb R^d$ from a distribution, the task is to find a unit vector $v$ that approximately maximizes the variance of the distribution after being projected along $v$. Despite being a classical task, standard estimators fail drastically if the data contains even a small fraction of outliers, motivating the problem of robust PCA. Recent work has developed computationally-efficient algorithms for robust PCA that either take super-linear time or have sub-optimal error guarantees. Our main contribution is to develop a nearly linear time algorithm for robust PCA with near-optimal error guarantees. We also develop a single-pass streaming algorithm for robust PCA with memory usage nearly-linear in the dimension.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-diakonikolas23a, title = {Nearly-Linear Time and Streaming Algorithms for Outlier-Robust {PCA}}, author = {Diakonikolas, Ilias and Kane, Daniel and Pensia, Ankit and Pittas, Thanasis}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {7886--7921}, 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/diakonikolas23a/diakonikolas23a.pdf}, url = {https://proceedings.mlr.press/v202/diakonikolas23a.html}, abstract = {We study principal component analysis (PCA), where given a dataset in $\mathbb R^d$ from a distribution, the task is to find a unit vector $v$ that approximately maximizes the variance of the distribution after being projected along $v$. Despite being a classical task, standard estimators fail drastically if the data contains even a small fraction of outliers, motivating the problem of robust PCA. Recent work has developed computationally-efficient algorithms for robust PCA that either take super-linear time or have sub-optimal error guarantees. Our main contribution is to develop a nearly linear time algorithm for robust PCA with near-optimal error guarantees. We also develop a single-pass streaming algorithm for robust PCA with memory usage nearly-linear in the dimension.} }
Endnote
%0 Conference Paper %T Nearly-Linear Time and Streaming Algorithms for Outlier-Robust PCA %A Ilias Diakonikolas %A Daniel Kane %A Ankit Pensia %A Thanasis Pittas %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-diakonikolas23a %I PMLR %P 7886--7921 %U https://proceedings.mlr.press/v202/diakonikolas23a.html %V 202 %X We study principal component analysis (PCA), where given a dataset in $\mathbb R^d$ from a distribution, the task is to find a unit vector $v$ that approximately maximizes the variance of the distribution after being projected along $v$. Despite being a classical task, standard estimators fail drastically if the data contains even a small fraction of outliers, motivating the problem of robust PCA. Recent work has developed computationally-efficient algorithms for robust PCA that either take super-linear time or have sub-optimal error guarantees. Our main contribution is to develop a nearly linear time algorithm for robust PCA with near-optimal error guarantees. We also develop a single-pass streaming algorithm for robust PCA with memory usage nearly-linear in the dimension.
APA
Diakonikolas, I., Kane, D., Pensia, A. & Pittas, T.. (2023). Nearly-Linear Time and Streaming Algorithms for Outlier-Robust PCA. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:7886-7921 Available from https://proceedings.mlr.press/v202/diakonikolas23a.html.

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