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.
|Title of host publication||30th Conference on Computational Complexity, CCC 2015|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|Number of pages||18|
|State||Published - 1 Jun 2015|
|Event||30th Conference on Computational Complexity, CCC 2015 - Portland, United States|
Duration: 17 Jun 2015 → 19 Jun 2015
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||30th Conference on Computational Complexity, CCC 2015|
|Period||17/06/15 → 19/06/15|
Bibliographical noteFunding Information:
Alex Samorodnitsky was supported by BSF and ISF grants. Ilya Shkredov was supported by Russian Scientific Foundation grant RSF 14-11-00433. We would like to thank Noga Alon, Swastik Kopparty, and Mark Rudelson for many helpful discussions regarding this work.
© A. Samorodnitsky and I. Shkredov and S. Yekhanin; licensed under Creative Commons License CC-BY.
- Kolmogorov width
- Linear codes
- Matrix rigidity