Year: 2024
Type: conference-paper
Abstract: 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.
Institution: University of California, Los Angeles
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4404848308