  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 $\omega$.

Computational Complexity;Discrete Mathematics