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Matrix discrepancy and the log-rank conjecture.

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Title: Matrix discrepancy and the log-rank conjecture.
Authors: Sudakov, Benny1 (AUTHOR) benjamin.sudakov@math.ethz.ch, Tomon, István2 (AUTHOR) istvan.tomon@umu.se
Source: Mathematical Programming. Jul2025, Vol. 212 Issue 1, p567-579. 13p.
Subjects: Semidefinite programming, Boolean functions, Logical prediction, Matrices (Mathematics)
Abstract: Given an m × n binary matrix M with | M | = p · m n (where |M| denotes the number of 1 entries), define the discrepancy of M as disc (M) = max X ⊂ [ m ] , Y ⊂ [ n ] | | M [ X × Y ] | - p | X | · | Y | | . Using semidefinite programming and spectral techniques, we prove that if rank (M) ≤ r and p ≤ 1 / 2 , then disc (M) ≥ Ω (m n) · min p , p 1 / 2 r . We use this result to obtain a modest improvement of Lovett's best known upper bound on the log-rank conjecture. We prove that any m × n binary matrix M of rank at most r contains an (m · 2 - O (r)) × (n · 2 - O (r)) sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank r is at most O (r) . [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Programming is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Mathematical+Programming%22">Mathematical Programming</searchLink>. Jul2025, Vol. 212 Issue 1, p567-579. 13p.
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  Data: Given an m × n binary matrix M with | M | = p · m n (where |M| denotes the number of 1 entries), define the discrepancy of M as disc (M) = max X ⊂ [ m ] , Y ⊂ [ n ] | | M [ X × Y ] | - p | X | · | Y | | . Using semidefinite programming and spectral techniques, we prove that if rank (M) ≤ r and p ≤ 1 / 2 , then disc (M) ≥ Ω (m n) · min p , p 1 / 2 r . We use this result to obtain a modest improvement of Lovett's best known upper bound on the log-rank conjecture. We prove that any m × n binary matrix M of rank at most r contains an (m · 2 - O (r)) × (n · 2 - O (r)) sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank r is at most O (r) . [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Mathematical Programming is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10107-024-02117-9
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        Text: English
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      – SubjectFull: Semidefinite programming
        Type: general
      – SubjectFull: Boolean functions
        Type: general
      – SubjectFull: Logical prediction
        Type: general
      – SubjectFull: Matrices (Mathematics)
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              Text: Jul2025
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