Sherman-Morrison-Woodbury Identity for Tensors

TL;DR

This paper extends the Sherman-Morrison-Woodbury identity from matrices to tensors, enabling efficient inverse updates and sensitivity analysis for multilinear systems, with theoretical derivations and numerical demonstrations.

Contribution

It derives the Sherman-Morrison-Woodbury identity for invertible tensors and generalizes it using Moore-Penrose inverse for non-invertible tensors, a novel extension in tensor algebra.

Findings

Derived the tensor Sherman-Morrison-Woodbury identity for invertible tensors.

Generalized the identity for tensors with Moore-Penrose inverse.

Numerical examples show the impact of perturbations on tensor inverse bounds.

Abstract

In linear algebra, the sherman-morrison-woodbury identity says that the inverse of a rank-$k$ correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix. This identity is crucial to accelerate the matrix inverse computation when the matrix involves correction. Many scientific and engineering applications have to deal with this matrix inverse problem after updating the matrix, e.g., sensitivity analysis of linear systems, covariance matrix update in kalman filter, etc. However, there is no similar identity in tensors. In this work, we will derive the sherman-morrison-woodbury identity for invertible tensors first. Since not all tensors are invertible, we further generalize the sherman-morrison-woodbury identity for tensors with moore-penrose generalized inverse by utilizing orthogonal projection of the correction tensor part into the original tensor and its Hermitian tensor. According to this new established the sherman-morrison-woodbury identity for tensors, we can perform sensitivity analysis for multi-linear systems by deriving the normalized upper bound for the solution of a multilinear system. Several numerical examples are also presented to demonstrate how the normalized error upper bounds are affected by perturbation degree of tensor coefficients.