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DOI: 10.4230/LIPIcs.CCC.2015.347

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

Authors: Alex Samorodnitsky, Ilya Shkredov, and Sergey Yekhanin

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

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.

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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)

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@InProceedings{samorodnitsky_et_al:LIPIcs.CCC.2015.347,
  author =	{Samorodnitsky, Alex and Shkredov, Ilya and Yekhanin, Sergey},
  title =	{{Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{347--364},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.347},
  URN =		{urn:nbn:de:0030-drops-50703},
  doi =		{10.4230/LIPIcs.CCC.2015.347},
  annote =	{Keywords: Matrix rigidity, linear codes, Kolmogorov width}
}

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DOI: 10.4230/DagSemProc.09421.4

Deterministic approximation algorithms for the nearest codeword problem

Authors: Noga Alon, Rina Panigrahy, and Sergey Yekhanin

Published in: Dagstuhl Seminar Proceedings, Volume 9421, Algebraic Methods in Computational Complexity (2010)

Abstract

The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v in F_2^n and a linear space L in F_2^n of dimension k NCP asks to find a point l in L that minimizes the (Hamming) distance from v. It is well-known that the nearest codeword problem is NP-hard. Therefore approximation algorithms are of interest. The best effcient approximation algorithms for the NCP to date are due to Berman and Karpinski. They are a deterministic algorithm that achieves an approximation ratio of O(k/c) for an arbitrary constant c; and a randomized algorithm that achieves an approximation ratio of O(k/ log n). In this paper we present new deterministic algorithms for approximating the NCP that improve substantially upon the earlier work, (almost) de-randomizing the randomized algorithm of Berman and Karpinski. We also initiate a study of the following Remote Point Problem (RPP). Given a linear space L in F_2^n of dimension k RPP asks to find a point v in F_2^n that is far from L. We say that an algorithm achieves a remoteness of r for the RPP if it always outputs a point v that is at least r-far from L. In this paper we present a deterministic polynomial time algorithm that achieves a remoteness of Omega(n log k / k) for all k < n/2. We motivate the remote point problem by relating it to both the nearest codeword problem and the matrix rigidity approach to circuit lower bounds in computational complexity theory.

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Noga Alon, Rina Panigrahy, and Sergey Yekhanin. Deterministic approximation algorithms for the nearest codeword problem. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{alon_et_al:DagSemProc.09421.4,
  author =	{Alon, Noga and Panigrahy, Rina and Yekhanin, Sergey},
  title =	{{Deterministic approximation algorithms for the nearest codeword problem}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--13},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{9421},
  editor =	{Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.4},
  URN =		{urn:nbn:de:0030-drops-24133},
  doi =		{10.4230/DagSemProc.09421.4},
  annote =	{Keywords: }
}

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

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Deterministic approximation algorithms for the nearest codeword problem

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