The Power of Sum-of-Squares for Detecting Hidden Structures

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 poly...

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Vydáno v:Annual Symposium on Foundations of Computer Science s. 720 - 731
Hlavní autoři: Hopkins, Samuel B., Kothari, Pravesh K., Potechin, Aaron, Raghavendra, Prasad, Schramm, Tselil, Steurer, David
Médium: Konferenční příspěvek
Jazyk:angličtina
Vydáno: IEEE 01.10.2017
Témata:
Algorithm design and analysis
average-case algorithms
average-case hardness
Inference algorithms
Mathematical model
Principal component analysis
Programming
Robustness
semidefinite programming
spectral algorithms
sum of squares
Tensile stress
ISSN:0272-5428
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Popis
Shrnutí: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.
ISSN:0272-5428
DOI:10.1109/FOCS.2017.72