[Paper Review] Tensor and Matrix Inversions with Applications

Michael Brazell, Na Li+2 more|arXiv (Cornell University)|Sep 18, 2011

Tensor decomposition and applications38 references3 citations

TL;DR

This paper establishes a group-theoretic framework for tensor inversion using the Einstein product, proving that even-order tensors form a group isomorphic to the general linear group. It enables multilinear system solving, tensor decompositions (CP and multilinear SVD), and numerical solutions via iterative methods like biconjugate gradient and Jacobi in tensor format, with applications to high-dimensional PDEs and quantum models.

ABSTRACT

Higher order tensor inversion is possible for even order. We have shown that a tensor group endowed with the Einstein (contracted) product is isomorphic to the general linear group of degree $n$. With the isomorphic group structures, we derived new tensor decompositions which we have shown to be related to the well-known canonical polyadic decomposition and multilinear SVD. Moreover, within this group structure framework, multilinear systems are derived, specifically, for solving high dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense, that is, when the tensor has an odd number of modes or when the tensor has distinct dimensions in each modes. With the notion of tensor inversion, multilinear systems are solvable. Numerically we solve multilinear systems using iterative techniques, namely biconjugate gradient and Jacobi methods in tensor format.

Motivation & Objective

• To establish a rigorous mathematical framework for tensor inversion using group theory and the Einstein product.
• To enable the solution of high-dimensional multilinear systems arising in PDEs and quantum mechanics.
• To derive tensor decompositions (CP and multilinear SVD) as special cases of group isomorphisms under symmetry constraints.
• To extend matrix-based numerical methods (e.g., biconjugate gradient, Jacobi) to tensor formats for iterative solution of multilinear systems.
• To address non-square and odd-mode tensors through least-squares formulations within the tensor group framework.

Proposed method

• Define a tensor group $(\mathbb{T}, \ast_2)$ under the Einstein product, showing it is isomorphic to the general linear group $\mathrm{GL}(n)$ via a bijective map $f$.
• Use the isomorphism $f: \mathbb{T} \to \mathbb{M}$ to transfer matrix group properties (associativity, identity, invertibility) to tensors.
• Define tensor inversion as $\widetilde{\mathcal{A}} = f^{-1}\left( [f(\mathcal{A})]^{-1} \right)$, ensuring $\widetilde{\mathcal{A}} \ast_2 \mathcal{A} = \mathcal{A} \ast_2 \widetilde{\mathcal{A}} = \mathscr{E}$, where $\mathscr{E}$ is the identity tensor with entries $\delta_{i_1 j_1} \delta_{i_2 j_2}$.
• Derive tensor decompositions (CP and multilinear SVD) as symmetric special cases of the group isomorphism, linking them to known tensor decompositions.
• Implement iterative solvers (biconjugate gradient, Jacobi) in tensor format to solve multilinear systems, particularly for high-dimensional PDEs and quantum models.
• Address non-invertible cases (odd-order or non-uniform mode dimensions) via least-squares minimization within the tensor group framework.

Experimental results

Research questions

• RQ1 Under what conditions is tensor inversion possible, and how can it be defined algebraically using group theory?
• RQ2 How can tensor decompositions such as CP and multilinear SVD be derived from isomorphic group structures?
• RQ3 Can iterative numerical methods be adapted to solve multilinear systems in tensor format for high-dimensional PDEs and quantum models?
• RQ4 How do the tensor group and isomorphism framework handle tensors with odd numbers of modes or distinct mode dimensions?
• RQ5 What is the role of the Einstein product in enabling multilinear systems and tensor inversion in a way analogous to matrix algebra?

Key findings

• The set of even-order tensors under the Einstein product forms a group isomorphic to the general linear group $\mathrm{GL}(n)$, enabling rigorous tensor inversion.
• The identity element for the tensor group is the tensor $\mathscr{E}$ with entries $\delta_{i_1 j_1} \delta_{i_2 j_2}$, which acts as the unit under the Einstein product.
• Tensor inversion is defined via the inverse of the corresponding matrix under the isomorphism $f$, ensuring $\widetilde{\mathcal{A}} \ast_2 \mathcal{A} = \mathcal{A} \ast_2 \widetilde{\mathcal{A}} = \mathscr{E}$.
• Tensor decompositions such as CP and multilinear SVD emerge as special cases of the group isomorphism when symmetries are imposed on the factors.
• Iterative solvers like biconjugate gradient and Jacobi are successfully adapted to tensor format for solving multilinear systems in high-dimensional PDEs and quantum models.
• The framework extends to non-invertible cases (odd-order or non-uniform mode tensors) through least-squares formulations, enabling numerical solution of over- or under-constrained multilinear systems.