Kolmogorov width of discrete linear spaces: An approach to matrix rigidity

Alex Samorodnitsky, Ilya Shkredov, Sergey Yekhanin

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 F2. Stated informally, we conjecture that under a suitable embedding of Fn2 into ℝn, vectors arising from a low dimensional F2-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.

Original languageAmerican English
Title of host publication30th Conference on Computational Complexity, CCC 2015
EditorsDavid Zuckerman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages347-364
Number of pages18
ISBN (Electronic)9783939897811
DOIs
StatePublished - 1 Jun 2015
Event30th Conference on Computational Complexity, CCC 2015 - Portland, United States
Duration: 17 Jun 201519 Jun 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume33
ISSN (Print)1868-8969

Conference

Conference30th Conference on Computational Complexity, CCC 2015
Country/TerritoryUnited States
CityPortland
Period17/06/1519/06/15

Bibliographical note

Publisher Copyright:
© A. Samorodnitsky and I. Shkredov and S. Yekhanin; licensed under Creative Commons License CC-BY.

Keywords

  • Kolmogorov width
  • Linear codes
  • Matrix rigidity

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