Matrix Multiplication and Number On the Forehead Communication

TL;DR

This paper reveals a deep connection between three-player Number On the Forehead communication complexity and matrix multiplication tensors, leveraging this link to transfer techniques and bounds between the two fields.

Contribution

It establishes that the Number On the Forehead communication problem corresponds exactly to the matrix multiplication tensor, enabling cross-application of methods and bounds.

Findings

Laser method can be adapted for communication protocols.

Lower bounds in communication imply bounds on matrix tensor properties.

Generalizes slice-rank methods to relate to matrix multiplication exponent .

Abstract

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 $\omega$.