The asymptotic spectrum of tensors and the exponent of matrix multiplication

@article{Strassen1986TheAS,
title={The asymptotic spectrum of tensors and the exponent of matrix multiplication},
author={Volker Strassen},
journal={27th Annual Symposium on Foundations of Computer Science (sfcs 1986)},
year={1986},
pages={49-54}
}
• V. Strassen
• Published 27 October 1986
• Computer Science, Mathematics
• 27th Annual Symposium on Foundations of Computer Science (sfcs 1986)
We introduce an asymptotic data structure for the relative bilinear complexity of bilinear maps (tensors). It consists of a compact Hausdorff space Δ together with an interpretation of the tensors under consideration as continuous functions on Δ. The asymptotic rank of a tensor is simply the maximum of the associated function. On the way we present a new method for estimating the exponent ω of matrix multiplication, leading at present to the bound ω ≪ 2.48. The paper gives only brief…
106 Citations
Universal points in the asymptotic spectrum of tensors
• Mathematics, Computer Science
STOC
• 2018
It is proved that the quantum functionals are asymptotic upper bounds on slice-rank and multi-slice rank, extending a result of Tao and Sawin and extending the Coppersmith–Winograd method via combinatorial degeneration.
Powers of tensors and fast matrix multiplication
This paper presents a method to analyze the powers of a given trilinear form and obtain upper bounds on the asymptotic complexity of matrix multiplication and obtains the upper bound ω < 2.3728639 on the exponent of square matrix multiplication, which slightly improves the best known upper bound.
Universal points in the asymptotic spectrum of tensors
• Mathematics
• 2018
The asymptotic restriction problem for tensors is to decide, given tensors s and t, whether the nth tensor power of s can be obtained from the (n+o(n))th tensor power of t by applying linear maps to
Powers of Tensors and Fast Matrix Multiplication
This paper presents a method to analyze the powers of a given linear form and obtain up to bounds on the asymptotic complexity of matrix multiplication and obtains the upper bound ω < 2.3728639 on the exponent of square matrix multiplication, which slightly improves the best known upper bound.
Algebraic complexity, asymptotic spectra and entanglement polytopes
The first explicit construction of an infinite family of elements in the asymptotic spectrum of complex k-tensors, based on information theory and entanglement polytopes is given, and it is proved that any polynomial can be efficiently approximated by a width-2 abp.
Asymptotic tensor rank of graph tensors: beyond matrix multiplication
• Computer Science, Mathematics
computational complexity
• 2018
An upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on k vertices is presented, showing that the exponent per edge is at most 0.77, outperforming the best known upper bound for matrix multiplication.
Symmetric Subrank of Tensors and Applications
• Mathematics
• 2021
Strassen (Strassen, J. Reine Angew. Math., 375/376, 1987) introduced the subrank of a tensor as a natural extension of matrix rank to tensors. Subrank measures the largest diagonal tensor that can be
The asymptotic spectrum of graphs and the Shannon capacity
A new dual characterisation of the Shannon capacity of graphs is obtained using the theory of asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) and elements in the asymPTotic spectrum of graphs include the Lovasz theta number, the fractional clique cover number, and the complement of the fractionale orthogonal rank.
Tensor rank is not multiplicative under the tensor product
• Mathematics, Computer Science
ArXiv
• 2017

References

SHOWING 1-10 OF 32 REFERENCES
Fast rectangular matrix multiplication (algebraic computational, asymptotic complexity)
The classical symmetrized construction of asymptotically fast algorithms that compute the product of two square matrices is generalized to construct algorithms that multiply certain rectangular matrices very efficiently.
Some Properties of Disjoint Sums of Tensors Related to Matrix Multiplication
Let t be a disjoint sum of tensors associated to matrix multiplication. The rank of the tensorial powers of t is bounded by an expression involving the elements of t and an exponent for matrix
On the Asymptotic Complexity of Matrix Multiplication
• Computer Science, Mathematics
SIAM J. Comput.
• 1982
A consequence of these results is that $\omega$, the exponent for matrix multiplication, is a limit point, that is, it cannot be realized by any single algorithm.
Relative bilinear complexity and matrix multiplication.
The significance of this notion lies, above all, in the key role of matrix multiplication for numerical linear algebra. Thus the following problems all have "exponent' : Matrix inversion,
The Complexity of Partial Derivatives
• Mathematics
Theor. Comput. Sci.
• 1983
Duality Applied to the Complexity of Matrix Multiplication and Other Bilinear Forms
• Mathematics, Computer Science
SIAM J. Comput.
• 1973
The paper considers the complexity of bilinear forms in a noncommutative ring. The dual of a computation is defined and applied to matrix multiplication and other bilinear forms. It is shown that t...
Partial and Total Matrix Multiplication
By combining Pan’s trilinear technique with a strong version of the compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of N \times N matrices (over arbitrary fields) is possible in time.
O(n2.7799) Complexity for n*n Approximate Matrix Multiplication
• Computer Science, Mathematics
Inf. Process. Lett.
• 1979
Evaluation of Rational Functions
• V. Strassen
• Mathematics
Complexity of Computer Computations
• 1972
In the first part of this paper the complexity (with respect to multiplication and division) of a general continued fraction and of arbitrary quadratic forms is determined with the help of Pan’s