The Communication Complexity of Approximating Matrix Rank

Alexander A. Sherstov; Andrey A. Storozhenko

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arXiv · 2410.20094

The Communication Complexity of Approximating Matrix Rank

Alexander A. Sherstov·Andrey A. Storozhenko

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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 $0\leq r 0$. 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 $S\cap T$).

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Alexander A. Sherstov, Andrey A. Storozhenko. 2024-10-26. The Communication Complexity of Approximating Matrix Rank. https://arxiv.org/abs/2410.20094

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