Matrix discrepancy and the log-rank conjecture.
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| Title: | Matrix discrepancy and the log-rank conjecture. |
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| 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] |
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| Database: | Engineering Source |
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| 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] |
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| ISSN: | 00255610 |
| DOI: | 10.1007/s10107-024-02117-9 |
