research-article

Rounding sum-of-squares relaxations

Authors:

Jonathan A. Kelner,

David SteurerAuthors Info & Claims

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

Pages 31 - 40

https://doi.org/10.1145/2591796.2591886

Published: 31 May 2014 Publication History

Abstract

We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof system to transform a combining algorithm---an algorithm that maps a distribution over solutions into a (possibly weaker) solution---into a rounding algorithm that maps a solution of the relaxation to a solution of the original problem.

Supplementary Material

MP4 File (p31-sidebyside.mp4)

Download
275.68 MB

References

[1]

A. Agarwal, A. Anandkumar, P. Jain, P. Netrapalli, and R. Tandon. Learning sparsely used overcomplete dictionaries via alternating minimization. arXiv preprint 1310.7991, 2013. http://arxiv.org/abs/1310.7991.

[2]

H.-C. An, R. Kleinberg, and D. B. Shmoys. Improving Christofides' algorithm for the s-t path TSP. In STOC, pages 875--886, 2012.

Digital Library

[3]

S. Arora, L. Babai, J. Stern, and Z. Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci., 54(2):317--331, 1997.

Digital Library

[4]

S. Arora, B. Barak, and D. Steurer. Subexponential algorithms for unique games and related problems. In FOCS, pages 563--572, 2010.

Digital Library

[5]

S. Arora, B. Bollobás, L. Lovász, and I. Tourlakis. Proving integrality gaps without knowing the linear program. Theory of Computing, 2(1):19--51, 2006. Preliminary version in FOCS '02.

[6]

S. Arora, R. Ge, and A. Moitra. New algorithms for learning incoherent and overcomplete dictionaries. arXiv preprint 1308.6723, 2013. http://arxiv.org/abs/1308.6273.

[7]

S. Arora, S. Rao, and U. V. Vazirani. Expander flows, geometric embeddings and graph partitioning. In STOC, pages 222--231, 2004.

Digital Library

[8]

B. Barak, F. G. S. L. Brandão, A. W. Harrow, J. A. Kelner, D. Steurer, and Y. Zhou. Hypercontractivity, sum-of-squares proofs, and their applications. In STOC, pages 307--326, 2012.

Digital Library

[9]

B. Barak, P. Gopalan, J. Håstad, R. Meka, P. Raghavendra, and D. Steurer. Making the long code shorter. In FOCS, pages 370--379, 2012.

Digital Library

[10]

B. Barak, J. Kelner, and D. Steurer. Iterative rounding for sum-of-squares relaxations. Preliminary version of the current work, unpublished, 2012.

[11]

B. Barak, J. Kelner, and D. Steurer. Rounding sum-of-squares relaxations. arXiv preprint arXiv:1312.6652, 2013. Full version of the current work.

[12]

B. Barak, J. Kelner, and D. Steurer. Dictionary learning via the sum-of-squares method. Manuscript in preparation, 2014.

[13]

B. Barak, P. Raghavendra, and D. Steurer. Rounding semidefinite programming hierarchies via global correlation. In FOCS, pages 472--481. IEEE, 2011.

Digital Library

[14]

S. Benabbas, K. Georgiou, A. Magen, and M. Tulsiani. SDP gaps from pairwise independence. Theory of Computing, 8(1):269--289, 2012.

[15]

A. Bhaskara, M. Charikar, A. Vijayaraghavan, V. Guruswami, and Y. Zhou. Polynomial integrality gaps for strong SDP relaxations of densest k-subgraph. In SODA, pages 388--405, 2012.

Digital Library

[16]

F. G. S. L. Brandão, M. Christandl, and J. Yard. A quasipolynomial-time algorithm for the quantum separability problem. In STOC, pages 343--352, 2011.

Digital Library

[17]

F. G. S. L. Brandão and A. W. Harrow. Quantum de Finetti theorems under local measurements with applications. In STOC, pages 861--870, 2013.

Digital Library

[18]

A. Bulatov, P. Jeavons, and A. Krokhin. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34(3):720--742, 2005. Preliminary version in ICALP '00.

Digital Library

[19]

M. Charikar, K. Makarychev, and Y. Makarychev. Integrality gaps for sherali-adams relaxations. In STOC, pages 283--292, 2009.

Digital Library

[20]

E. Chlamtac and M. Tulsiani. Convex relaxations and integrality gaps, 2010. Chapter in Handbook on Semidefinite, Cone and Polynomial Optimization.

[21]

W. F. de la Vega and C. Kenyon-Mathieu. Linear programming relaxations of maxcut. In Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, pages 53--61. Society for Industrial and Applied Mathematics, 2007.

Digital Library

[22]

L. Demanet and P. Hand. Recovering the Sparsest Element in a Subspace, Oct. 2013. Arxiv preprint 1310.1654.

[23]

A. C. Doherty and S. Wehner. Convergence of SDP hierarchies for polynomial optimization on the hypersphere. arXiv preprint arXiv:1210.5048, 2012.

[24]

S. O. Gharan, A. Saberi, and M. Singh. A randomized rounding approach to the traveling salesman problem. In FOCS, pages 550--559, 2011.

Digital Library

[25]

M. X. Goemans and D. P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115--1145, 1995. Preliminary version in STOC '94.

Digital Library

[26]

L.-A. Gottlieb and T. Neylon. Matrix sparsification and the sparse null space problem. In APPROX-RANDOM, pages 205--218, 2010.

Digital Library

[27]

D. Grigoriev. Complexity of Positivstellensatz proofs for the knapsack. Computational Complexity, 10(2):139--154, 2001.

Digital Library

[28]

D. Grigoriev. Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity. Theor. Comput. Sci., 259(1-2):613--622, 2001.

Digital Library

[29]

D. Grigoriev and N. Vorobjov. Complexity of Null-and Positivstellensatz proofs. Annals of Pure and Applied Logic, 113(1):153--160, 2001.

[30]

V. Guruswami and A. K. Sinop. Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with psd objectives. In FOCS, pages 482--491, 2011.

Digital Library

[31]

A. W. Harrow and A. Montanaro. Testing product states, quantum merlin-arthur games and tensor optimization. J. ACM, 60(1):3, 2013. Preliminary version in FOCS '10.

Digital Library

[32]

M. Kauers, R. O'Donnell, L.-Y. Tan, and Y. Zhou. Hypercontractive inequalities via sos, and the frankl-rödl graph. In SODA, 2014.

Digital Library

[33]

S. Khot. On the power of unique 2-prover 1-round games. In IEEE Conference on Computational Complexity, page 25, 2002.

Digital Library

[34]

S. Khot and N. K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into &ell;₁. In FOCS, pages 53--62, 2005.

Digital Library

[35]

P. Koiran and A. Zouzias. Hidden cliques and the certification of the restricted isometry property. CoRR, abs/1211.0665, 2012.

[36]

J.-L. Krivine. Anneaux préordonnés. Journal d'analyse mathématique, 12(1):307--326, 1964.

[37]

J. B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11(3):796--817, 2001.

Digital Library

[38]

M. Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging applications of algebraic geometry, pages 157--270. Springer, 2009. Updated version available on the author's homepage at http://homepages.cwi.nl/~monique/files/moment-ima-update-new.pdf.

[39]

L. Lovász and A. Schrijver. Cones of matrices and set-functions and 0-1 optimization. SIAM Journal on Optimization, 1(2):166--190, 1991.

Digital Library

[40]

Y. Nesterov. Squared functional systems and optimization problems. High performance optimization, 13:405--440, 2000.

[41]

R. O'Donnell and Y. Zhou. Approximability and proof complexity. In SODA, pages 1537--1556, 2013.

Digital Library

[42]

P. A. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, 2000.

[43]

P. Raghavendra. Complexity of constraint satisfaction problems: Exact and approximate, 2010. Talk at the Institute for Advanced Study, video available on http://video.ias.edu/csdm/complexityconstraint.

[44]

P. Raghavendra and D. Steurer. Graph expansion and the unique games conjecture. In STOC, pages 755--764, 2010.

Digital Library

[45]

P. Raghavendra, D. Steurer, and P. Tetali. Approximations for the isoperimetric and spectral profile of graphs and related parameters. In STOC, pages 631--640, 2010.

Digital Library

[46]

P. Raghavendra, D. Steurer, and M. Tulsiani. Reductions between expansion problems. In IEEE Conference on Computational Complexity, pages 64--73, 2012.

Digital Library

[47]

G. Schoenebeck. Linear level Lasserre lower bounds for certain k-CSPs. In FOCS, pages 593--602, 2008.

Digital Library

[48]

H. D. Sherali and W. P. Adams. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics, 3(3):411--430, 1990.

Digital Library

[49]

N. Shor. An approach to obtaining global extremums in polynomial mathematical programming problems. Cybernetics and Systems Analysis, 23(5):695--700, 1987.

[50]

D. A. Spielman, H. Wang, and J. Wright. Exact recovery of sparsely-used dictionaries. Journal of Machine Learning Research - Proceedings Track, 23:37.1--37.18, 2012.

[51]

G. Stengle. A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. Mathematische Annalen, 207(2):87--97, 1974.

[52]

M. Tulsiani. CSP gaps and reductions in the Lasserre hierarchy. In STOC, pages 303--312, 2009.

Digital Library

[53]

M. Zibulevsky and B. A. Pearlmutter. Blind source separation by sparse decomposition in a signal dictionary. Neural computation, 13(4):863--882, 2001.

Digital Library

Cited By

Guruswami VHsieh JRaghavendra P(2024)Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond the $\sqrt{n}$ Dimension Threshold2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00062(930-948)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00062
Liu AMoitra A(2023)Robustly Learning General Mixtures of GaussiansJournal of the ACM10.1145/358368070:3(1-53)Online publication date: 23-May-2023
https://dl.acm.org/doi/10.1145/3583680
Chen SKoehler FMoitra AYau MLeonardi SGupta A(2022)Kalman filtering with adversarial corruptionsProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520050(832-845)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520050
Show More Cited By

Index Terms

Rounding sum-of-squares relaxations
1. Theory of computation
  1. Design and analysis of algorithms

Recommendations

Conic mixed-integer rounding cuts

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined ...
Sum-of-Squares Relaxations in Robust DC Optimization and Feature Selection
Abstract
This paper presents sum-of-squares (SOS) relaxation results to a difference-of-convex-max (DC-max) optimization involving SOS-convex polynomials in the face of constraint data uncertainty and their applications to robust feature selection. The ...
Polyhedral properties of RLT relaxations of nonconvex quadratic programs and their implications on exact relaxations: Polyhedral properties of RLT relaxations of nonconvex quadratic...
Abstract
We study linear programming relaxations of nonconvex quadratic programs given by the reformulation–linearization technique (RLT), referred to as RLT relaxations. We investigate the relations between the polyhedral properties of the feasible ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

May 2014

984 pages

ISBN:9781450327107

DOI:10.1145/2591796

Program Chair:
David Shmoys
Cornell University

Copyright © 2014 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 31 May 2014

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Research-article

Conference

STOC '14

Sponsor:

SIGACT

STOC '14: Symposium on Theory of Computing

May 31 - June 3, 2014

New York, New York

Acceptance Rates

STOC '14 Paper Acceptance Rate 91 of 319 submissions, 29%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Upcoming Conference

STOC '25

Sponsor:
sigact

57th Annual ACM Symposium on Theory of Computing (STOC 2025)

June 23 - 27, 2025

Prague , Czech Republic

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

29
Total Citations
View Citations
391
Total Downloads

Downloads (Last 12 months)23
Downloads (Last 6 weeks)6

Reflects downloads up to 08 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Guruswami VHsieh JRaghavendra P(2024)Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond the $\sqrt{n}$ Dimension Threshold2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00062(930-948)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00062
Liu AMoitra A(2023)Robustly Learning General Mixtures of GaussiansJournal of the ACM10.1145/358368070:3(1-53)Online publication date: 23-May-2023
https://dl.acm.org/doi/10.1145/3583680
Chen SKoehler FMoitra AYau MLeonardi SGupta A(2022)Kalman filtering with adversarial corruptionsProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520050(832-845)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520050
Barak BMoitra A(2022)Noisy tensor completion via the sum-of-squares hierarchyMathematical Programming10.1007/s10107-022-01793-9193:2(513-548)Online publication date: 29-Mar-2022
https://doi.org/10.1007/s10107-022-01793-9
Bafna MBarak BKothari PSchramm TSteurer DKhuller SVassilevska Williams V(2021)Playing unique games on certified small-set expandersProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451099(1629-1642)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451099
Liu AMoitra AKhuller SVassilevska Williams V(2021)Settling the robust learnability of mixtures of GaussiansProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451084(518-531)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451084
Ding YKunisky DWein ABandeira A(2021)The Average-Case Time Complexity of Certifying the Restricted Isometry PropertyIEEE Transactions on Information Theory10.1109/TIT.2021.311282367:11(7355-7361)Online publication date: Nov-2021
https://doi.org/10.1109/TIT.2021.3112823
Bogdanov ASabin MVasudevan PChan T(2019)XOR codes and sparse learning parity with noiseProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310496(986-1004)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310496
Krajíček J(2019)Proof Complexity10.1017/9781108242066Online publication date: 25-Mar-2019
https://doi.org/10.1017/9781108242066
Hopkins SKothari PPotechin ARaghavendra PSchramm T(2018)On the Integrality Gap of Degree-4 Sum of Squares for Planted CliqueACM Transactions on Algorithms10.1145/317853814:3(1-31)Online publication date: 16-Jun-2018
https://dl.acm.org/doi/10.1145/3178538
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten