Name
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
<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup>
16]), we prove a new SoS lower bound for the tensor PCA problem. |
Year | DOI | Venue |
|---|---|---|
2017 | 10.1109/FOCS.2017.72 | 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) |
Keywords | DocType | Volume |
sum of squares,semidefinite programming,average-case algorithms,average-case hardness,spectral algorithms | Journal | abs/1710.05017 |
ISSN | ISBN | Citations |
0272-5428 | 978-1-5386-3465-3 | 6 |
PageRank | References | Authors |
0.41 | 26 | 6 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Samuel Hopkins | 1 | 88 | 9.47 |
| Pravesh Kothari | 2 | 203 | 22.17 |
| Aaron Potechin | 3 | 63 | 7.81 |
| Prasad Raghavendra | 4 | 1013 | 50.58 |
| Tselil Schramm | 5 | 78 | 8.82 |
| David Steurer | 6 | 934 | 44.91 |