The Communication Complexity of Approximating Matrix Rank

TL;DR

This paper determines the exact randomized and quantum communication complexity of approximating matrix rank over finite fields, providing tight bounds and implications for streaming algorithms and other linear algebra problems.

Contribution

It establishes the first strong lower bounds for approximating matrix rank in the general case, extending previous results and applying to quantum protocols and error probabilities.

Findings

Communication complexity is (1 + r^2 \u2217 \u2212log|\u211d|) for rank approximation.

Optimality of bounds is shown with matching upper bounds and quantum protocol considerations.

Lower bounds imply space complexity limits for streaming algorithms approximating matrix rank.

Abstract

We fully determine the communication complexity of approximating matrix rank, over any finite field $\mathbb{F}$. We study the most general version of this problem, where $0\leq r<R\leq n$ are given integers, Alice and Bob's inputs are matrices $A,B\in\mathbb{F}^{n\times n}$, respectively, and they need to distinguish between the cases $\mathrm{rk}(A+B)=r$ and $\mathrm{rk}(A+B)=R$. We show that this problem has randomized communication complexity $\Omega(1+r^{2}\log|\mathbb{F}|)$. This is optimal in a strong sense because $O(1+r^{2}\log|\mathbb{F}|)$ communication is sufficient to determine, for arbitrary $A,B$, whether $\mathrm{rk}(A+B)\leq r$. Prior to our work, lower bounds were known only for consecutive integers $r$ and $R$, with no implication for the approximation of matrix rank. Our lower bound holds even for quantum protocols and even for error probability $\frac{1}{2}-\frac{1}{4}|\mathbb{F}|^{-r/3}$, which too is virtually optimal because the problem has a two-bit classical protocol with error $\frac{1}{2}-\Theta(|\mathbb{F}|^{-r})$. As an application, we obtain an $\Omega(\frac{1}{k}\cdot n^{2}\log|\mathbb{F}|)$ space 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. We also settle the randomized and quantum communication complexity of several other linear-algebraic problems, for all settings of parameters. This includes the determinant problem (given matrices $A$ and $B$, distinguish between the cases $\mathrm{det}(A+B)=a$ and $\mathrm{det}(A+B)=b$, for fixed field elements $a\ne b)$ and the subspace sum and subspace intersection problem (given subspaces $S$ and $T$ of known dimensions $m$ and $\ell$, respectively, approximate the dimensions of $S+T$ and $S\cap T$).