A Polynomial Formulation for Joint Decomposition of Symmetric Tensors of Different Orders

Tensors with eigenvectors in a given subspace

Rendiconti del Circolo Matematico di Palermo (1952 -) ◽

10.1007/s12215-021-00600-2 ◽

2021 ◽

Author(s):

Giorgio Ottaviani ◽

Zahra Shahidi

Keyword(s):

Algebraic Geometry ◽

Linear Subspace ◽

Chern Classes ◽

Symmetric Tensors

AbstractThe first author with B. Sturmfels studied in [16] the variety of matrices with eigenvectors in a given linear subspace, called the Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the irreducibility of the Kalman variety and computing its codimension and degree. Furthermore, we consider the Kalman variety of tensors having singular t-tuples with the first component in a given linear subspace and we prove analogous results, which are new even in the case of matrices. Main techniques come from Algebraic Geometry, using Chern classes for enumerative computations.

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A nonmonotone accelerated Levenberg–Marquardt method for the ‐eigenvalues of symmetric tensors

International Transactions in Operational Research ◽

10.1111/itor.12954 ◽

2021 ◽

Author(s):

Mingyuan Cao ◽

Yueting Yang ◽

Tianliang Hou ◽

Chaoqian Li

Keyword(s):

Symmetric Tensors ◽

Marquardt Method ◽

Levenberg Marquardt

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Fast Bilinear Algorithms for Symmetric Tensor Contractions

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2019-0075 ◽

2020 ◽

Vol 0 (0) ◽

Cited By ~ 1

Author(s):

Edgar Solomonik ◽

James Demmel

Keyword(s):

Symmetric Matrix ◽

Matrix Multiplication ◽

Computational Cost ◽

Symmetric Tensor ◽

Partial Sums ◽

Symmetric Tensors ◽

Outer Product ◽

Bilinear Complexity ◽

Special Cases ◽

Tensor Contractions

AbstractIn 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 {\frac{(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\times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

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