The Communication Complexity of Approximating Matrix Rank

We fully determine the communication complexity of approximating matrix rank, over any finite field F. We study the most general version of this problem, where $0\leqslant r < R\leqslant n$ are given integers and Alice and Bob need to determine whether their respective matrices $A, B\in \mathbb{F}^{n\times n}$ satisfy rk $(A+B)=r$ versus rk $(A+B)=R$. We show that this problem has commu-nication cost $\Omega(r^{2}\log\vert \mathbb{F}\vert)$, which is optimal. Our lower bound holds even for quantum protocols and even for error probability $\frac{1}{2}(1-\vert \mathbb{F}\vert ^{-r/3})$, which too is optimal because this problem has a two-bit classical protocol with error $\frac{1}{2}(1-\vert \mathbb{F}\vert ^{-\Theta(r)})$. Prior to our work, lower bounds were known only for constant-error protocols and only for consecutive integers $r$ and $R$, with no implication for the approximation of matrix rank. We also settle an analogous question for subspaces, where Alice has a subspace $S$, Bob has a subspace $T$) and they need to approximate the dimension of the subspace $S+T$ generated by $S$ and $T$ (equivalently, approximate the dimension of $S\cap T$). As an application, we obtain an $\Omega(n^{2}\log\vert \mathbb{F}\vert)/k$ memory lower bound for any streaming algorithm with $k$ passes that approximates the rank of an input matrix $M\in \mathbb{F}^{n\times n}$ within a factor of $\sqrt{2}-\delta$, for any $\delta > 0$. Our result is an exponential improvement in $k$ over previous work.