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Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity

Authors Alex Samorodnitsky, Ilya Shkredov, Sergey Yekhanin



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Alex Samorodnitsky
Ilya Shkredov
Sergey Yekhanin

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Alex Samorodnitsky, Ilya Shkredov, and Sergey Yekhanin. Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 347-364, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.CCC.2015.347

Abstract

A square matrix V is called rigid if every matrix V' obtained by altering a small number of entries of $V$ has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major implications in computational complexity theory. One approach to establishing rigidity of a matrix V is to come up with a property that is satisfied by any collection of vectors arising from a low-dimensional space, but is not satisfied by the rows of V even after alterations. In this paper we propose such a candidate property that has the potential of establishing rigidity of combinatorial design matrices over the field F_2. Stated informally, we conjecture that under a suitable embedding of F_2^n into R^n, vectors arising from a low dimensional F_2-linear space always have somewhat small Kolmogorov width, i.e., admit a non-trivial simultaneous approximation by a low dimensional Euclidean space. This implies rigidity of combinatorial designs, as their rows do not admit such an approximation even after alterations. Our main technical contribution is a collection of results establishing weaker forms and special cases of the conjecture above.
Keywords
  • Matrix rigidity
  • linear codes
  • Kolmogorov width

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References

  1. Michael Alekhnovich. More on average case vs. approximation complexity. In 44th IEEE Symposium on Foundations of Computer Science (FOCS), pages 298-307, 2003. Google Scholar
  2. Noga Alon and Gil Cohen. On rigid matrices and U-polynomials. In 28th IEEE Computational Complexity Conference (CCC), pages 197-206, 2013. Google Scholar
  3. Noga Alon, Rina Panigrahy, and Sergey Yekhanin. Deterministic approximation algorithms for the nearest codeword problem. In 13th International Workshop on Randomization and Computation (RANDOM), volume 5687 of Lecture Notes in Computer Science, pages 339-351, 2009. Google Scholar
  4. Zhidong Bai and Jack Silverstein. Spectral Analysis of Large Dimensional Random Matrices. Springer, New York, 2010. Google Scholar
  5. Zeev Dvir. On matrix rigidity and locally self-correctable codes. Computational Complexity, 20:367-388, 2011. Google Scholar
  6. Joel Friedman. A note on matrix rigidity. Combinatorica, 13:235-239, 1993. Google Scholar
  7. Noboru Hamada. On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes. Hiroshima Mathematical Journal, 3:154-226, 1973. Google Scholar
  8. R. S. Ismagilov. n-dimensional width of compact in Hilbert space. Journal of Functional Analysis and Applications, 2:32-39, 1968. Google Scholar
  9. Stasys Jukna. Extremal combinatorics. Springer, Berlin, Heidelberg, New York, 2001. Google Scholar
  10. Dieter Jungnickel and Vladimir Tonchev. Polarities, quasi-symmetric designs, and hamada conjecture. Designs, Codes, and Cryptography, 51:131-140, 2009. Google Scholar
  11. Boris Kashin and Alexander Razborov. Improved lower bounds on the rigidity of Hadamard matrices. Mathematical Notes, 63:471-475, 1998. Google Scholar
  12. Satyanarayana Lokam. Spectral methods for matrix rigidity with applications to size-depth trade-offs and communication complexity. Journal of Computer and System Sciences, 63:449-473, 2001. Google Scholar
  13. Satyanarayana Lokam. Complexity lower bounds using linear algebra. Foundations and Trends in Theoretical Computer Science, 4:1-155, 2009. Google Scholar
  14. Ilan Newman and Yuri Rabinovich. On multiplicative λ-approximations and some geometric applications. SIAM Journal on Computing, 42:885-883, 2013. Google Scholar
  15. Alexander Razborov. On rigid matrices, 1989. Manuscript. In Russian. Google Scholar
  16. Alexander Razborov and Steven Rudich. Natural proofs. Journal of Computer and System Sciences, 55:24-35, 1997. Google Scholar
  17. Shubhangi Saraf and Sergey Yekhanin. Noisy interpolation of sparse polynomials, and applications. In 26th IEEE Computational Complexity Conference (CCC), pages 86-92, 2011. Google Scholar
  18. Rocco Servedio and Emanuele Viola. On a special case of rigidity. In Electronic Colloquium on Computational Complexity (ECCC), TR12-144, 2012. Google Scholar
  19. Amin Shokrollahi, Daniel Speilman, and Voelker Stemann. A remark on matrix rigidity. Information Processing Letters, 64:283-285, 1997. Google Scholar
  20. Kempton Smith. On the p-rank of the incidence matrix of points and hyperplanes in a finite projective geometry. Journal of Combinatorial Theory, 7:122-129, 1969. Google Scholar
  21. Vladimir Temlyakov. Nonlinear Kolmogorov widths. Mathematical Notes, 63:785-795, 1998. Google Scholar
  22. Kirill Uskov. Kolmogorov width of geometric configurations and functional compacts in Hilbert spaces. PhD thesis, Moscow State Technical University, 2002. in Russian. Google Scholar
  23. Leslie Valiant. Graph-theoretic arguments in low level complexity. In 6th Symposium on Mathematical Foundations of Computer Science (MFCS), pages 162-176, 1977. Google Scholar
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