Matrix discrepancy and the log-rank conjecture

Abstract

AbstractGiven an $$m\times n$$ m × n binary matrix M with $$|M|=p\cdot mn$$ | M | = p · m n (where |M| denotes the number of 1 entries), define the discrepancy of M as $${{\,\textrm{disc}\,}}(M)=\displaystyle \max \nolimits _{X\subset [m], Y\subset [n]}\big ||M[X\times Y]|-p|X|\cdot |Y|\big |$$ disc ( M ) = max X ⊂ [ m ] , Y ⊂ [ n ] | | M [ X × Y ] | - p | X | · | Y | | . Using semidefinite programming and spectral techniques, we prove that if $${{\,\textrm{rank}\,}}(M)\le r$$ rank ( M ) ≤ r and $$p\le 1/2$$ p ≤ 1 / 2 , then $$\begin{aligned}{{\,\textrm{disc}\,}}(M)\ge \Omega (mn)\cdot \min \left\{ p,\frac{p^{1/2}}{\sqrt{r}}\right\} .\end{aligned}$$ 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 $$m\times n$$ m × n binary matrix M of rank at most r contains an $$(m\cdot 2^{-O(\sqrt{r})})\times (n\cdot 2^{-O(\sqrt{r})})$$ ( 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 $$O(\sqrt{r})$$ O ( r ) .