The power of sum-of-squares for detecting hidden structures

Samuel B. Hopkins, Pravesh K. Kothari, Aaron Potechin, Prasad Raghavendra, Tselil Schramm, David Steurer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

14 Scopus citations

Abstract

We study planted problems-finding hidden structures in random noisy inputs-through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new algorithms for planted problems, often achieving the best known polynomial-time guarantees in terms of accuracy of recovered solutions and robustness to noise. One theme in recent work is the design of spectral algorithms which match the guarantees of SoS algorithms for planted problems. Classical spectral algorithms are often unable to accomplish this: The twist in these new spectral algorithms is the use of spectral structure of matrices whose entries are low-degree polynomials of the input variables.We prove that for a wide class of planted problems, including refuting random constraint satisfaction problems, tensor and sparse PCA, densest-ksubgraph, community detection in stochastic block models, planted clique, and others, eigenvalues of degree-d matrix polynomials are as powerful as SoS semidefinite programs of degree d. For such problems it is therefore always possible to match the guarantees of SoS without solving a large semidefinite program.Using related ideas on SoS algorithms and lowdegree matrix polynomials (and inspired by recent work on SoS and the planted clique problem [BHK+16]), we prove a new SoS lower bound for the tensor PCA problem.

Original languageEnglish (US)
Title of host publicationProceedings - 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017
PublisherIEEE Computer Society
Pages720-731
Number of pages12
ISBN (Electronic)9781538634646
DOIs
StatePublished - Nov 10 2017
Event58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017 - Berkeley, United States
Duration: Oct 15 2017Oct 17 2017

Publication series

NameAnnual Symposium on Foundations of Computer Science - Proceedings
Volume2017-October
ISSN (Print)0272-5428

Other

Other58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017
CountryUnited States
CityBerkeley
Period10/15/1710/17/17

All Science Journal Classification (ASJC) codes

  • Computer Science(all)

Keywords

  • average-case algorithms
  • average-case hardness
  • semidefinite programming
  • spectral algorithms
  • sum of squares

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  • Cite this

    Hopkins, S. B., Kothari, P. K., Potechin, A., Raghavendra, P., Schramm, T., & Steurer, D. (2017). The power of sum-of-squares for detecting hidden structures. In Proceedings - 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017 (pp. 720-731). [8104104] (Annual Symposium on Foundations of Computer Science - Proceedings; Vol. 2017-October). IEEE Computer Society. https://doi.org/10.1109/FOCS.2017.72