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Resilience of the rank of random matrices

Part of: Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  28 August 2020

Asaf Ferber ,
Kyle Luh  and
Gweneth McKinley
Asaf Ferber
Affiliation:
Department of Mathematics, University of California, Irvine, CA92697, USA
Kyle Luh
Affiliation:
Department of Mathematics, University of Colorado Boulder, Campus Box 395, 2300 Colorado Avenue, Boulder, CO80309-0395, USA
Gweneth McKinley*
Affiliation:
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA92093-0112, USA
*
*Corresponding author. Email: gmckinley@ucsd.edu
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Abstract

Let M be an n × m matrix of independent Rademacher (±1) random variables. It is well known that if $n \leq m$, then M is of full rank with high probability. We show that this property is resilient to adversarial changes to M. More precisely, if $m \ge n + {n^{1 - \varepsilon /6}}$, then even after changing the sign of (1 – ε)m/2 entries, M is still of full rank with high probability. Note that this is asymptotically best possible as one can easily make any two rows proportional with at most m/2 changes. Moreover, this theorem gives an asymptotic solution to a slightly weakened version of a conjecture made by Van Vu in [17].

MSC classification

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see )
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 2 , March 2021 , pp. 163 - 174
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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