TY - JOUR
T1 - Fast bilinear algorithms for symmetric tensor contractions
AU - Solomonik, Edgar
AU - Demmel, James
N1 - Funding Information:
This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. Via XSEDE, the authors made use of the TACC Stampede2 supercomputer.
Publisher Copyright:
© 2021 De Gruyter. All rights reserved.
PY - 2021/1/1
Y1 - 2021/1/1
N2 - In matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order s + v and v + t by a factor of (s+t+v)!/s!t!v! to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58× for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.
AB - In matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order s + v and v + t by a factor of (s+t+v)!/s!t!v! to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58× for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.
KW - Bilinear Complexity
KW - Symmetric Matrices
KW - Symmetric Tensors
KW - Tensor Contractions
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U2 - 10.1515/cmam-2019-0075
DO - 10.1515/cmam-2019-0075
M3 - Article
AN - SCOPUS:85079619179
VL - 21
SP - 211
EP - 231
JO - Computational Methods in Applied Mathematics
JF - Computational Methods in Applied Mathematics
SN - 1609-4840
IS - 1
ER -