### Abstract

We study the computational complexity of approximating the 2-to-q norm of linear operators (defined as ∥A∥ _{2→q} = max _{v≠0}∥Av| _{q}/∥v∥ _{2}) for q > 2, as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following: 1. For any constant even integer q ≥ 4, a graph G is a small-set expander if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2→q norm. As a corollary, a good approximation to the 2→q norm will refute the Small-Set Expansion Conjecture - - a close variant of the UGC. We also show that such a good approximation can be obtained in exp(n ^{2/q}) time, thus obtaining a different proof of the known subexponential algorithm for Small-Set-Expansion. 2. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy certify an upper bound on the 2→4 norm of the projector to low degree polynomials over the Boolean cube, as well certify the unsatisfiability of the "noisy cube" and "short code" based instances of Unique-Games considered by prior works. This improves on the previous upper bound of exp(log ^{O(1)} n) rounds (for the "short code"), as well as separates the "Sum of Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to require ω(1) rounds. 3. We show reductions between computing the 2→4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory. Three corollaries are: (i) the 2→4 norm is NP-hard to approximate to precision inverse-polynomial in the dimension, (ii) the 2→4 norm does not have a good approximation (in the sense above) unless 3-SAT can be solved in time exp(√n poly log(n)), and (iii) known algorithms for the quantum separability problem imply a non-trivial additive approximation for the 2→4 norm.

Original language | English (US) |
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Title of host publication | STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing |

Pages | 307-326 |

Number of pages | 20 |

DOIs | |

State | Published - Jun 26 2012 |

Externally published | Yes |

Event | 44th Annual ACM Symposium on Theory of Computing, STOC '12 - New York, NY, United States Duration: May 19 2012 → May 22 2012 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Other

Other | 44th Annual ACM Symposium on Theory of Computing, STOC '12 |
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Country | United States |

City | New York, NY |

Period | 5/19/12 → 5/22/12 |

### Fingerprint

### Keywords

- hypercontractive
- injective tensor norm
- semidefinite programming
- unique games conjecture

### ASJC Scopus subject areas

- Software

### Cite this

*STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing*(pp. 307-326). (Proceedings of the Annual ACM Symposium on Theory of Computing). https://doi.org/10.1145/2213977.2214006

**Hypercontractivity, sum-of-squares proofs, and their applications.** / Barak, Boaz; Brandão, Fernando G.S.L.; Harrow, Aram W.; Kelner, Jonathan; Steurer, David; Zhou, Yuan.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing.*Proceedings of the Annual ACM Symposium on Theory of Computing, pp. 307-326, 44th Annual ACM Symposium on Theory of Computing, STOC '12, New York, NY, United States, 5/19/12. https://doi.org/10.1145/2213977.2214006

}

TY - GEN

T1 - Hypercontractivity, sum-of-squares proofs, and their applications

AU - Barak, Boaz

AU - Brandão, Fernando G.S.L.

AU - Harrow, Aram W.

AU - Kelner, Jonathan

AU - Steurer, David

AU - Zhou, Yuan

PY - 2012/6/26

Y1 - 2012/6/26

N2 - We study the computational complexity of approximating the 2-to-q norm of linear operators (defined as ∥A∥ 2→q = max v≠0∥Av| q/∥v∥ 2) for q > 2, as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following: 1. For any constant even integer q ≥ 4, a graph G is a small-set expander if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2→q norm. As a corollary, a good approximation to the 2→q norm will refute the Small-Set Expansion Conjecture - - a close variant of the UGC. We also show that such a good approximation can be obtained in exp(n 2/q) time, thus obtaining a different proof of the known subexponential algorithm for Small-Set-Expansion. 2. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy certify an upper bound on the 2→4 norm of the projector to low degree polynomials over the Boolean cube, as well certify the unsatisfiability of the "noisy cube" and "short code" based instances of Unique-Games considered by prior works. This improves on the previous upper bound of exp(log O(1) n) rounds (for the "short code"), as well as separates the "Sum of Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to require ω(1) rounds. 3. We show reductions between computing the 2→4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory. Three corollaries are: (i) the 2→4 norm is NP-hard to approximate to precision inverse-polynomial in the dimension, (ii) the 2→4 norm does not have a good approximation (in the sense above) unless 3-SAT can be solved in time exp(√n poly log(n)), and (iii) known algorithms for the quantum separability problem imply a non-trivial additive approximation for the 2→4 norm.

AB - We study the computational complexity of approximating the 2-to-q norm of linear operators (defined as ∥A∥ 2→q = max v≠0∥Av| q/∥v∥ 2) for q > 2, as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following: 1. For any constant even integer q ≥ 4, a graph G is a small-set expander if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2→q norm. As a corollary, a good approximation to the 2→q norm will refute the Small-Set Expansion Conjecture - - a close variant of the UGC. We also show that such a good approximation can be obtained in exp(n 2/q) time, thus obtaining a different proof of the known subexponential algorithm for Small-Set-Expansion. 2. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy certify an upper bound on the 2→4 norm of the projector to low degree polynomials over the Boolean cube, as well certify the unsatisfiability of the "noisy cube" and "short code" based instances of Unique-Games considered by prior works. This improves on the previous upper bound of exp(log O(1) n) rounds (for the "short code"), as well as separates the "Sum of Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to require ω(1) rounds. 3. We show reductions between computing the 2→4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory. Three corollaries are: (i) the 2→4 norm is NP-hard to approximate to precision inverse-polynomial in the dimension, (ii) the 2→4 norm does not have a good approximation (in the sense above) unless 3-SAT can be solved in time exp(√n poly log(n)), and (iii) known algorithms for the quantum separability problem imply a non-trivial additive approximation for the 2→4 norm.

KW - hypercontractive

KW - injective tensor norm

KW - semidefinite programming

KW - unique games conjecture

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U2 - 10.1145/2213977.2214006

DO - 10.1145/2213977.2214006

M3 - Conference contribution

AN - SCOPUS:84862597933

SN - 9781450312455

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 307

EP - 326

BT - STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing

ER -