On Rigid Matrices and U-Polynomials
We introduce a class of polynomials, which we call U - polynomials , and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of U -pol...
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| Vydáno v: | Computational complexity Ročník 24; číslo 4; s. 851 - 879 |
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Springer International Publishing
01.12.2015
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| Abstract | We introduce a class of polynomials, which we call
U
-
polynomials
, and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of
U
-polynomials, though their size is larger than desired. Furthermore, we give two alternative proofs for the fact that small-bias sets induce rigid matrices.
Finally, we construct rigid matrices from unbalanced expanders, with essentially the same size as the construction via small-bias sets. |
|---|---|
| AbstractList | We introduce a class of polynomials, which we call
U
-
polynomials
, and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of
U
-polynomials, though their size is larger than desired. Furthermore, we give two alternative proofs for the fact that small-bias sets induce rigid matrices.
Finally, we construct rigid matrices from unbalanced expanders, with essentially the same size as the construction via small-bias sets. |
| Author | Cohen, Gil Alon, Noga |
| Author_xml | – sequence: 1 givenname: Noga surname: Alon fullname: Alon, Noga organization: Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University – sequence: 2 givenname: Gil surname: Cohen fullname: Cohen, Gil email: gil.cohen@weizmann.ac.il organization: Department of Computer Science and Applied Mathematics, Weizmann Institute of Science |
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| Cites_doi | 10.1145/1538902.1538904 10.1002/rsa.3240050203 10.1109/18.119713 10.1002/rsa.3240030308 10.1007/978-3-642-03685-9_26 10.1090/S0273-0979-06-01126-8 10.1109/CCC.2010.35 10.1109/SFCS.1995.492457 10.1016/0012-365X(88)90189-6 10.1016/S0020-0190(97)00190-7 10.1017/CBO9781139814782 10.1007/BF02311250 10.1137/0222053 10.1007/3-540-08353-7_135 10.1007/BF01303207 10.1109/FOCS.2009.44 |
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| Keywords | small-bias sets Matrix rigidity unbalanced expanders 68Q17 68R05 |
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| References_xml | – reference: PudlákP.RödlV.Pseudorandom sets and explicit constructions of Ramsey graphsQuad. Mat200413327346 – reference: N. Alon, J. Bruck, J. Naor, M. Naor & R. Roth (1992a). Construction of Asymptotically Good Low-Rate Error-Correcting Codes through Pseudo-Random Graphs. IEEE Transactions on Information Theory38, 509–516. – reference: L. G. Valiant (1977). Graph-theoretic arguments in low-level complexity. In Lecture notes in Computer Science, volume 53, 162–176. Springer. – reference: FriedmanJ.A note on matrix rigidityCombinatorica19931322352390848.15005123704510.1007/BF01303207 – reference: NaorJ.NaorM.Small-Bias Probability Spaces: Efficient Constructions and ApplicationsSIAM J. on Computing19932248388560776.60014122776410.1137/0222053 – reference: LokamS.V.Complexity Lower Bounds using Linear AlgebraFoundations and Trends in Theoretical Computer Science200941-211551176.680842539154 – reference: A. Ben-Aroya & A. Ta-Shma (2009). Constructing small-bias sets from algebraic-geometric codes. In Proceedings of the 50th Annual IEEE symposium on Foundations of Computer science (FOCS), 191–197. – reference: F. J. MacWilliams & N. J. A. Sloane (1977). The Theory of Error-Correcting Codes, Part II. North-Holland. – reference: ShokrollahiM.A.SpielmanD.StemannV.A remark on matrix rigidityInformation Processing Letters1997646283285160824010.1016/S0020-0190(97)00190-7 – reference: AlonN.GoldreichO.HåstadJ.PeraltaR.Simple Construction of Almost k-wise Independent Random VariablesRandom Structures and Algorithms1992332893040755.60002116484210.1002/rsa.3240030308 – reference: AlonN.RoichmanY.Random Cayley Graphs and ExpandersRandom Structures and Algorithms1994522712850798.05048126297910.1002/rsa.3240050203 – reference: Z. Dvir (2010). On Matrix Rigidity and Locally Self-Correctable Codes. In Proceedings of the 25th IEEE Conference on Computational Complexity (CCC), 291–298. – reference: V. Arvind & S. Srinivasan (2010). The Remote Point Problem, Small Bias Spaces, and Expanding Generator Sets. In Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS), 59–70. – reference: HooryS.LinialN.WigdersonA.Expander Graphs and their ApplicationsBulleting of the American Mathematical Society2006434395611147.68608224791910.1090/S0273-0979-06-01126-8 – reference: S. V. Lokam (1995). Spectral Methods for Matrix Rigidity with Applications to Size-Depth Tradeoffs and Communication Complexity. In Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS), 6–15. – reference: AlonN.ChungF.R.K.Explicit Construction of Linear Sized Tolerant NetworksDiscrete Mathematics198872115190657.0506897551910.1016/0012-365X(88)90189-6 – reference: GuruswamiV.UmansC.VadhanS.Unbalanced expanders and randomness extractors from Parvaresh–Vardy codesJ. ACM2009564134259082210.1145/1538902.1538904 – reference: KashinB.S.RazborovA.A.Improved lower bounds on the rigidity of Hadamard matricesMathematical Notes19986344714750917.15013168094310.1007/BF02311250 – reference: N. Alon, R. Panigrahy & S. Yekhanin (2009). Deterministic approximation algorithms for the nearest codeword problem. In APPROX 09 / RANDOM 09: Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 339–351. – reference: R. O’Donnell (2014). Analysis of boolean functions. Cambridge University Press. – volume: 56 start-page: 1 issue: 4 year: 2009 ident: 112_CR10 publication-title: J. ACM doi: 10.1145/1538902.1538904 – volume: 4 start-page: 1 issue: 1-2 year: 2009 ident: 112_CR14 publication-title: Foundations and Trends in Theoretical Computer Science – volume: 5 start-page: 271 issue: 2 year: 1994 ident: 112_CR5 publication-title: Random Structures and Algorithms doi: 10.1002/rsa.3240050203 – ident: 112_CR1 doi: 10.1109/18.119713 – volume: 3 start-page: 289 issue: 3 year: 1992 ident: 112_CR3 publication-title: Random Structures and Algorithms doi: 10.1002/rsa.3240030308 – ident: 112_CR4 doi: 10.1007/978-3-642-03685-9_26 – volume: 43 start-page: 439 year: 2006 ident: 112_CR11 publication-title: Bulleting of the American Mathematical Society doi: 10.1090/S0273-0979-06-01126-8 – ident: 112_CR8 doi: 10.1109/CCC.2010.35 – ident: 112_CR13 doi: 10.1109/SFCS.1995.492457 – volume: 72 start-page: 15 issue: 1 year: 1988 ident: 112_CR2 publication-title: Discrete Mathematics doi: 10.1016/0012-365X(88)90189-6 – ident: 112_CR15 – volume: 13 start-page: 327 year: 2004 ident: 112_CR18 publication-title: Quad. Mat – volume: 64 start-page: 283 issue: 6 year: 1997 ident: 112_CR19 publication-title: Information Processing Letters doi: 10.1016/S0020-0190(97)00190-7 – ident: 112_CR17 doi: 10.1017/CBO9781139814782 – ident: 112_CR6 – volume: 63 start-page: 471 issue: 4 year: 1998 ident: 112_CR12 publication-title: Mathematical Notes doi: 10.1007/BF02311250 – volume: 22 start-page: 838 issue: 4 year: 1993 ident: 112_CR16 publication-title: SIAM J. on Computing doi: 10.1137/0222053 – ident: 112_CR20 doi: 10.1007/3-540-08353-7_135 – volume: 13 start-page: 235 issue: 2 year: 1993 ident: 112_CR9 publication-title: Combinatorica doi: 10.1007/BF01303207 – ident: 112_CR7 doi: 10.1109/FOCS.2009.44 |
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U
-
polynomials
, and show that the problem of explicitly constructing a rigid matrix can be reduced to the... |
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| Title | On Rigid Matrices and U-Polynomials |
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