SIAM Journal on Matrix Analysis and Applications· 2013Q1
Solving Multilinear Systems via Tensor Inversion
- 214citations
- Q1SCImago
- 2013year
Short summary
A novel tensor inversion method is developed for solving multilinear systems, applicable to even-order tensors and equivalent to canonical polyadic decomposition under certain constraints.
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Key points
- Developed a tensor inversion method for solving multilinear systems, applicable to even-order tensors.
- Established an isomorphism between tensor groups and the general linear group, enabling tensor SVD.
- Tensor SVD is shown to be equivalent to canonical polyadic decomposition and multilinear SVD under constraints.
- Applied the method to solve problems in high-dimensional PDEs and large discrete quantum models.
- Iterative methods (biconjugate gradient, Jacobi) are used for multilinear systems outside the main framework.
AI-generated from the title and abstract; the full text is not read.
Abstract
Higher order tensor inversion is possible for even order. This is due to the fact that a tensor group endowed with the contracted product is isomorphic to the general linear group of degree $n$. With these isomorphic group structures, we derive a tensor SVD which we have shown to be equivalent to well-known canonical polyadic decomposition and multilinear SVD provided that some constraints are satisfied. Moreover, within this group structure framework, multilinear systems are derived and solved for problems of high-dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense. These are cases when there is an odd number of modes or when each mode has distinct dimension. Numerically we solve multilinear systems using iterative techniques, namely, biconjugate gradient and Jacobi methods.
The authors' abstract, as published at the source. SIAM Journal on Matrix Analysis and Applications, 2013 · DOI ↗
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Field: Computational Mathematics
Computational MathematicsMathematics