### 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 language | English (US) |
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Title of host publication | Proceedings - 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017 |

Publisher | IEEE Computer Society |

Pages | 720-731 |

Number of pages | 12 |

ISBN (Electronic) | 9781538634646 |

DOIs | |

State | Published - Nov 10 2017 |

Event | 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017 - Berkeley, United States Duration: Oct 15 2017 → Oct 17 2017 |

### Publication series

Name | Annual Symposium on Foundations of Computer Science - Proceedings |
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Volume | 2017-October |

ISSN (Print) | 0272-5428 |

### Other

Other | 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017 |
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Country | United States |

City | Berkeley |

Period | 10/15/17 → 10/17/17 |

### 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

*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