Three-player Number On the Forehead communication may be thought of as a three-player Number In the Hand promise model, in which each player is given the inputs that are supposedly on the other two players' heads, and promised that they are consistent with the inputs of of the other players. The set of all allowed inputs under this promise may be thought of as an order-3 tensor. We surprisingly observe that this tensor is exactly the matrix multiplication tensor, which is widely studied in the design of fast matrix multiplication algorithms. Using this connection, we prove a number of results about both Number On the Forehead communication and matrix multiplication, each by using known results or techniques about the other. For example, we show how the Laser method, a key technique used to design the best matrix multiplication algorithms, can also be used to design communication protocols for a variety of problems. We also show how known lower bounds for Number On the Forehead communication can be used to bound properties of the matrix multiplication tensor such as its zeroing out subrank. Finally, we substantially generalize known methods based on slice-rank for studying communication, and show how they directly relate to the matrix multiplication exponent $\omega$.

翻译：三人“前额数字”（Number On the Forehead）通信可视为一种三人“手中数字”（Number In the Hand）承诺模型，其中每位玩家被赋予理应位于另两位玩家前额上的输入，并承诺这些输入与其他玩家的输入一致。在此承诺下，所有允许的输入集合可视为一个三阶张量。我们惊人地观察到，该张量恰好是矩阵乘法张量，后者在快速矩阵乘法算法的设计中已被广泛研究。利用这一关联，我们通过借鉴对方领域的已知结果或技术，证明了关于“前额数字”通信与矩阵乘法的多项结论。例如，我们展示了用于设计最优矩阵乘法算法的关键技术——激光方法（Laser method），同样可用于设计多种问题的通信协议。我们还表明，已知的“前额数字”通信下界可用来界定矩阵乘法张量的性质，如消零子秩（zeroing out subrank）。最后，我们显著推广了基于切片秩（slice-rank）研究通信的已知方法，并直接揭示了它们与矩阵乘法指数 $\omega$ 之间的联系。