dc.contributor.author | Barak, Boaz | |
dc.contributor.author | Brandao, Fernando G.S.L. | |
dc.contributor.author | Harrow, Aram W. | |
dc.contributor.author | Steurer, David | |
dc.contributor.author | Zhou, Yuan | |
dc.contributor.author | Kelner, Jonathan Adam | |
dc.date.accessioned | 2013-09-11T14:35:27Z | |
dc.date.available | 2013-09-11T14:35:27Z | |
dc.date.issued | 2012-05 | |
dc.identifier.isbn | 9781450312455 | |
dc.identifier.uri | http://hdl.handle.net/1721.1/80386 | |
dc.description | Original manuscript October 30, 2012 | en_US |
dc.description.abstract | We study the computational complexity of approximating the 2-to-q norm of linear operators (defined as |A|[subscript 2->q] = max[subscript v≠ 0]|Av|[subscript q]/|v|[subscript 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: 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[superscript 2/q]) time, thus obtaining a different proof of the known subexponential algorithm for Small-Set-Expansion. 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[superscript 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. 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. | en_US |
dc.description.sponsorship | National Science Foundation (U.S.) (Grant 0916400) | en_US |
dc.description.sponsorship | National Science Foundation (U.S.) (Grant 0829937) | en_US |
dc.description.sponsorship | United States. Defense Advanced Research Projects Agency. Quantum Entanglement Science and Technology (Contract FA9550-09-1-0044) | en_US |
dc.description.sponsorship | National Science Foundation (U.S.) (Award 1111109) | en_US |
dc.description.sponsorship | National Science Foundation (U.S.) (Award 0843915) | en_US |
dc.description.sponsorship | United States. Intelligence Advanced Research Projects Activity (Quantum Computer Science) | en_US |
dc.language.iso | en_US | |
dc.publisher | Association for Computing Machinery (ACM) | en_US |
dc.relation.isversionof | http://dx.doi.org/10.1145/2213977.2214006 | en_US |
dc.rights | Creative Commons Attribution-Noncommercial-Share Alike 3.0 | en_US |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-sa/3.0/ | en_US |
dc.source | arXiv | en_US |
dc.title | Hypercontractivity, sum-of-squares proofs, and their applications | en_US |
dc.type | Article | en_US |
dc.identifier.citation | Boaz Barak, Fernando G.S.L. Brandao, Aram W. Harrow, Jonathan Kelner, David Steurer, and Yuan Zhou. 2012. Hypercontractivity, sum-of-squares proofs, and their applications. In Proceedings of the 44th symposium on Theory of Computing (STOC '12). ACM, New York, NY, USA, 307-326. | en_US |
dc.contributor.department | Massachusetts Institute of Technology. Department of Mathematics | en_US |
dc.contributor.department | Massachusetts Institute of Technology. Department of Physics | en_US |
dc.contributor.department | McGovern Institute for Brain Research at MIT | en_US |
dc.contributor.mitauthor | Barak, Boaz | en_US |
dc.contributor.mitauthor | Harrow, Aram W. | en_US |
dc.contributor.mitauthor | Kelner, Jonathan Adam | en_US |
dc.relation.journal | Proceedings of the 44th symposium on Theory of Computing (STOC '12) | en_US |
dc.eprint.version | Original manuscript | en_US |
dc.type.uri | http://purl.org/eprint/type/JournalArticle | en_US |
eprint.status | http://purl.org/eprint/status/NonPeerReviewed | en_US |
dspace.orderedauthors | Barak, Boaz; Brandao, Fernando G.S.L.; Harrow, Aram W.; Kelner, Jonathan; Steurer, David; Zhou, Yuan | en_US |
dc.identifier.orcid | https://orcid.org/0000-0002-4257-4198 | |
dc.identifier.orcid | https://orcid.org/0000-0003-3220-7682 | |
dc.identifier.orcid | https://orcid.org/0000-0002-4120-4048 | |
mit.license | OPEN_ACCESS_POLICY | en_US |
mit.metadata.status | Complete | |