Complexity Theoretic Lower Bounds for Sparse Principal Component Detection

Quentin Berthet, Philippe Rigollet
Proceedings of the 26th Annual Conference on Learning Theory, PMLR 30:1046-1066, 2013.

Abstract

In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite programming. We also prove that the statistical performance of this test cannot be strictly improved by any computationally efficient method. Our results can be viewed as complexity theoretic lower bounds conditionally on the assumptions that some instances of the planted clique problem cannot be solved in randomized polynomial time.

Cite this Paper

BibTeX
@InProceedings{pmlr-v30-Berthet13, title = {Complexity Theoretic Lower Bounds for Sparse Principal Component Detection}, author = {Berthet, Quentin and Rigollet, Philippe}, booktitle = {Proceedings of the 26th Annual Conference on Learning Theory}, pages = {1046--1066}, year = {2013}, editor = {Shalev-Shwartz, Shai and Steinwart, Ingo}, volume = {30}, series = {Proceedings of Machine Learning Research}, address = {Princeton, NJ, USA}, month = {12--14 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v30/Berthet13.pdf}, url = {https://proceedings.mlr.press/v30/Berthet13.html}, abstract = {In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite programming. We also prove that the statistical performance of this test cannot be strictly improved by any computationally efficient method. Our results can be viewed as complexity theoretic lower bounds conditionally on the assumptions that some instances of the planted clique problem cannot be solved in randomized polynomial time. } }
Endnote
%0 Conference Paper %T Complexity Theoretic Lower Bounds for Sparse Principal Component Detection %A Quentin Berthet %A Philippe Rigollet %B Proceedings of the 26th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2013 %E Shai Shalev-Shwartz %E Ingo Steinwart %F pmlr-v30-Berthet13 %I PMLR %P 1046--1066 %U https://proceedings.mlr.press/v30/Berthet13.html %V 30 %X In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite programming. We also prove that the statistical performance of this test cannot be strictly improved by any computationally efficient method. Our results can be viewed as complexity theoretic lower bounds conditionally on the assumptions that some instances of the planted clique problem cannot be solved in randomized polynomial time.
RIS
TY - CPAPER TI - Complexity Theoretic Lower Bounds for Sparse Principal Component Detection AU - Quentin Berthet AU - Philippe Rigollet BT - Proceedings of the 26th Annual Conference on Learning Theory DA - 2013/06/13 ED - Shai Shalev-Shwartz ED - Ingo Steinwart ID - pmlr-v30-Berthet13 PB - PMLR DP - Proceedings of Machine Learning Research VL - 30 SP - 1046 EP - 1066 L1 - http://proceedings.mlr.press/v30/Berthet13.pdf UR - https://proceedings.mlr.press/v30/Berthet13.html AB - In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite programming. We also prove that the statistical performance of this test cannot be strictly improved by any computationally efficient method. Our results can be viewed as complexity theoretic lower bounds conditionally on the assumptions that some instances of the planted clique problem cannot be solved in randomized polynomial time. ER -
APA
Berthet, Q. & Rigollet, P.. (2013). Complexity Theoretic Lower Bounds for Sparse Principal Component Detection. Proceedings of the 26th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 30:1046-1066 Available from https://proceedings.mlr.press/v30/Berthet13.html.

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