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Perturbation analysis of tensor $(\mathcal{B},\mathcal{C})$-inverse via Einstein product

Daochang Zhang, Jingqian Li, Dijana Mosic, Predrag S. Stanimirovic

TL;DR

This work develops a perturbation theory for tensor generalized inverses under the Einstein product, focusing on how small perturbations $\mathcal{E}$ of a tensor $\mathcal{A}$ influence inner, outer, and $(\mathcal{B},\mathcal{C})$-inverses. Using multiplicative correction factors $\rho$, $\delta$, and their variants, the authors derive precise conditions and formulas that relate perturbed inverses $(\mathcal{A}+\mathcal{E})^{(1)}$, $(\mathcal{A}+\mathcal{E})^{(2)}$, and $(\mathcal{A}+\mathcal{E})^{(2)}_{\text{R}(\mathcal{B}),\text{N}(\mathcal{C})}$ to their unperturbed counterparts, along with explicit error bounds. A key contribution is the tensor analogue of perturbation results from Banach spaces, including a Finite Rshrank Theorem that ties perturbation validity to preserved reshape rank, and comprehensive perturbation formulas for inner inverses and for outer inverses with prescribed ranges and/or null spaces. These results yield stable computational strategies for tensor inverses in applications involving the Einstein product, with practical error estimates and algorithmic implications for numerical tensor analysis.

Abstract

We investigate the influence of a relatively small perturbation on various generalized inverses functions or quantities derived from a tensor $\mathcal{A}$.When a small tensor perturbation \(\mathcal{E}\) is introduced, it becomes challenging to analyze generalized inverses of the perturbed tensor \( \mathcal{D} =\mathcal{A}+\mathcal{E}\) and to determine how this perturbation affects a generalized inverse of $\mathcal{A}$.Our main goal is to understand the relationship between $\mathcal{D}^\Game$ and \( \mathcal{A}^\Game \), where $(\cdot)^\Game$ denotes a specific generalized inverse or a class of generalized inverses.In particular, classes of tensor inner, outer, and $(\mathcal{B},\mathcal{C})$ inverses are considered.

Perturbation analysis of tensor $(\mathcal{B},\mathcal{C})$-inverse via Einstein product

TL;DR

This work develops a perturbation theory for tensor generalized inverses under the Einstein product, focusing on how small perturbations of a tensor influence inner, outer, and -inverses. Using multiplicative correction factors , , and their variants, the authors derive precise conditions and formulas that relate perturbed inverses , , and to their unperturbed counterparts, along with explicit error bounds. A key contribution is the tensor analogue of perturbation results from Banach spaces, including a Finite Rshrank Theorem that ties perturbation validity to preserved reshape rank, and comprehensive perturbation formulas for inner inverses and for outer inverses with prescribed ranges and/or null spaces. These results yield stable computational strategies for tensor inverses in applications involving the Einstein product, with practical error estimates and algorithmic implications for numerical tensor analysis.

Abstract

We investigate the influence of a relatively small perturbation on various generalized inverses functions or quantities derived from a tensor .When a small tensor perturbation is introduced, it becomes challenging to analyze generalized inverses of the perturbed tensor and to determine how this perturbation affects a generalized inverse of .Our main goal is to understand the relationship between and , where denotes a specific generalized inverse or a class of generalized inverses.In particular, classes of tensor inner, outer, and inverses are considered.
Paper Structure (5 sections, 22 theorems, 109 equations, 1 algorithm)

This paper contains 5 sections, 22 theorems, 109 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Key lemma
  3. Perturbation of inner inverse
  4. Perturbation of outer and $(\mathcal{B},\mathcal{C})$-inverses
  5. Conclusion

Key Result

Lemma 2.3

LMA685 Let $\mathcal{A} \in \mathbb{C}^{S(s) \times T(t)}$, $\mathcal{X} \in \mathbb{C}^{T(t) \times S(s)}$, $\mathcal{B} \in \mathbb{C}^{T(t) \times U(u)}$, $C \in \mathbb{C}^{V(v) \times S(s)}$. The tensor $\mathcal{A}_{\aunclfamily{\bf R}(\mathcal{B}),*}^{(2)}$ is an outer inverse of $\mathcal{A} and

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 30 more