Folded Polynomial Codes for Coded Distributed $AA^\top$-Type Matrix Multiplication

TL;DR

This paper introduces folded polynomial codes for distributed $AA^ op$ matrix multiplication, improving efficiency and straggler mitigation in distributed systems by achieving optimal recovery thresholds.

Contribution

The paper proposes a novel folded polynomial code strategy that outperforms existing methods in distributed matrix multiplication, especially for $AA^ op$ computations.

Findings

Achieves the optimal recovery threshold for $m=1$ cases.

Outperforms existing strategies in recovery threshold, download cost, and decoding complexity.

Provides a lower bound on the recovery threshold for linear strategies over real numbers.

Abstract

In this paper, due to the important value in practical applications, we consider the coded distributed matrix multiplication problem of computing $AA^\top$ in a distributed computing system with $N$ worker nodes and a master node, where the input matrices $A$ and $A^\top$ are partitioned into $m$-by-$p$ and $p$-by-$m$ blocks of equal-size sub-matrices respectively. For effective straggler mitigation, we propose a novel computation strategy, named \emph{folded polynomial code}, which is obtained by modifying the entangled polynomial codes. Moreover, we characterize a lower bound on the optimal recovery threshold among all linear computation strategies when the underlying field is the real number field, and our folded polynomial codes can achieve this bound in the case of $m=1$. Compared with all known computation strategies for coded distributed matrix multiplication, our folded polynomial codes outperform them in terms of recovery threshold, download cost, and decoding complexity.