We provide novel coded computation strategies for distributed matrix–matrix products that outperform the recent “Polynomial code” constructions in recovery threshold, i.e., the required number of successful workers. When a fixed $1/m$ fraction of each matrix can be stored at each worker node, Polynomial codes require $m^{2}$ successful workers, while our MatDot codes only require $2m-1$ successful workers. However, MatDot codes have higher computation cost per worker and higher communication cost from each worker to the fusion node. We also provide a systematic construction of MatDot codes. Furthermore, we propose “PolyDot” coding that interpolates between Polynomial codes and MatDot codes to trade off computation/communication costs and recovery thresholds. Finally, we demonstrate a novel coding technique for multiplying $n$ matrices ( $n \geq 3$ ) using ideas from MatDot and PolyDot codes.

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关于编码矩阵乘法的最优恢复阈值

我们为分布式矩阵-矩阵产品提供了新颖的编码计算策略，其在恢复阈值（即所需的成功工人数量）方面优于最近的“多项式代码”结构。当一个固定 1 美元/百万美元 每个矩阵的分数可以存储在每个工作节点，多项式代码需要 $m^{2}$ 成功的工人，而我们的 MatDot 代码只需要 200 万-1 美元 成功的工人。然而，MatDot 代码每个工人的计算成本更高，每个工人到融合节点的通信成本也更高。我们还提供了 MatDot 代码的系统构建。此外，我们提出了在多项式代码和 MatDot 代码之间进行插值的“PolyDot”编码，以权衡计算/通信成本和恢复阈值。最后，我们展示了一种新的乘法编码技术 $n$ 矩阵（ $n \geq 3$ ) 使用来自 MatDot 和 PolyDot 代码的想法。