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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.54

The Full Rank Condition for Sparse Random Matrices

Authors: Amin Coja-Oghlan, Jane Gao, Max Hahn-Klimroth, Joon Lee, Noela Müller, and Maurice Rolvien

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. Inspired by low-density parity check codes, the family of random matrices that we investigate is very general and encompasses both matrices over finite fields and {0,1}-matrices over the rationals. The proof combines statistical physics-inspired coupling techniques with local limit arguments.

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Amin Coja-Oghlan, Jane Gao, Max Hahn-Klimroth, Joon Lee, Noela Müller, and Maurice Rolvien. The Full Rank Condition for Sparse Random Matrices. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cojaoghlan_et_al:LIPIcs.APPROX/RANDOM.2023.54,
  author =	{Coja-Oghlan, Amin and Gao, Jane and Hahn-Klimroth, Max and Lee, Joon and M\"{u}ller, Noela and Rolvien, Maurice},
  title =	{{The Full Rank Condition for Sparse Random Matrices}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{54:1--54:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.54},
  URN =		{urn:nbn:de:0030-drops-188792},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.54},
  annote =	{Keywords: random matrices, rank, finite fields, rationals}
}

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The Full Rank Condition for Sparse Random Matrices

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