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Matrix Multiplication

Tensors represent matrix multiplication of fixed dimensions: . Modes represent input and output matrices. The first mode corresponds to the first input matrix A with entries in row-major order; the second mode corresponds to the second input matrix B with entries in row-major order; the third mode corresponds to the output matrix C with entries in column-major order. The choice of orderings reveals the cyclic symmetry of the tensor. The tensor is binary valued. Nonzero entries (ones) correspond to scalar multiplications within the classical matrix multiplication algorithm: if entry then the th entry of A is multiplied by the th entry of B and is accumulated into the th entry of C.

The naming convention specifies the three dimensions . The ordering of the dimensions is irrelevant, as all 6 possible tensors are equivalent under simple transformations.

The exact rank of these tensors corresponds to the optimal bilinear complexity of matrix multiplication for the specified dimensions. The border rank of the tensor corresponds to an optimal arbitrary-precision-approximation (APA) algorithm.

Tensor Statistics

Non-zeros M*K*N
Order 3
Dimensions M*K x K*N x M*N
Tags binary , cyclic symmetric

Downloadable Files

File Description
matmul_2-2-2.tns.gz Tensor
matmul_3-3-3.tns.gz Tensor
matmul_4-3-2.tns.gz Tensor
matmul_4-4-3.tns.gz Tensor
matmul_4-4-4.tns.gz Tensor
matmul_5-5-5.tns.gz Tensor
matmul_6-3-3.tns.gz Tensor
matmul_generator.m Tensor generator (Matlab)


  Title                    = {A Framework for Practical Parallel Fast Matrix Multiplication},
  Author                   = {Benson, Austin R. and Ballard, Grey},
  Booktitle                = {Proceedings of the 20th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming},
  Year                     = {2015},
  Pages                    = {42--53},
  Series                   = {PPoPP 2015},
  Doi                      = {10.1145/2688500.2688513},
  Url                      = {}
  Title                    = {Algorithms for matrix multiplication},
  Author                   = {Brent, Richard P.},
  Institution              = {Stanford University},
  Year                     = {1970},