Real tensor factorizations and generalized inverses under the $t$-product

Technical Summary: Real Tensor Factorizations and Generalized Inverses under the t-Product

Problem Statement
The algebraic theory of third-order tensors under the t-product is traditionally formulated over the complex field $\mathbb{C}$ via Fourier block diagonalization. While this approach allows tensor operations to be reduced to matrix operations on frontal slices in the transform domain, many practical applications (e.g., image processing, video analysis, hyperspectral imaging) inherently involve real-valued data. A critical gap exists in ensuring that tensor factorizations and generalized inverses constructed in the complex Fourier domain yield real-valued tensors after inverse transformation. The paper addresses the structural conditions required to guarantee that these transform-domain constructions result in real tensors, noting that simply performing blockwise operations on complex Fourier blocks does not automatically preserve real-valuedness.

Methodology
The paper adopts a framework based on the discrete Fourier transform (DFT) along the third dimension of a tensor. Key methodological components include:

1. Fourier Block Diagonalization: Utilizing the isomorphism between the t-product ring and block-circulant matrices, which are unitarily similar to block-diagonal matrices containing the Fourier transforms of the frontal slices.
2. Conjugate-Pairing Principle: The core theoretical mechanism introduced is the "conjugate-pairing principle." The authors establish that for a tensor to be real-valued, its Fourier blocks must satisfy specific conjugate symmetry relations ($A_{p-k+2} = \overline{A_k}$), rather than requiring each individual block to be real.
3. Compatible Construction: The methodology involves performing matrix-level decompositions (SVD, Schur, Jordan, etc.) on the Fourier blocks. Crucially, these decompositions are not chosen independently for each block. Instead, the authors enforce compatibility across conjugate pairs of blocks to ensure the resulting inverse transform yields a real tensor. This is formalized in Lemma 3.1, which proves that if a matrix operation $\Phi$ satisfies $\Phi(\overline{M}) = \overline{\Phi(M)}$, applying it to conjugate-symmetric Fourier blocks preserves real-valuedness.

Key Contributions and Results

• Structural Characterization: The paper provides necessary and sufficient conditions (Lemmas 2.4 and 2.5) for a tensor to be real-valued based on the conjugate symmetry of its Fourier blocks.
• Real Tensor Factorizations: The authors derive real versions of several fundamental tensor decompositions by applying the conjugate-pairing principle:
  • Real t-SVD: A decomposition $A = U * S * V^T$ where $U$ and $V$ are orthogonal tensors and $S$ is f-diagonal. The paper demonstrates that independent SVDs of Fourier blocks fail to produce real factors, necessitating the coupling of singular vectors across conjugate pairs.
  • Real t-Schur Decomposition: A decomposition $A = U * T * U^T$ where $T$ is an f-quasi-triangular tensor (frontal slices are quasi-triangular with $1\times1$ and $2\times2$ diagonal blocks), analogous to the real Schur form for matrices.
  • Real t-Jordan Canonical Form: A decomposition $A = P * J * P^{-1}$ where $J$ is an f-upper-block-bi-diagonal tensor. The paper corrects a previous assertion in the literature that the t-Jordan form of a real tensor is automatically real, showing instead that a specific conjugate-pairing of Jordan data is required.
  • Real Idempotent Factorization: A factorization $A = U * E * V$ where $E$ is a real idempotent tensor.
• Generalized Inverses: The framework is applied to generalized inverses, proving the existence and providing transform-domain descriptions for:
  • Moore–Penrose Inverse: $A^\dagger = V * S^\dagger * U^T$, shown to be real-valued.
  • Drazin Inverse: $A^D$ exists as a real tensor if the Fourier blocks admit Drazin inverses satisfying the pairing condition.
  • Group Inverse: $A^\#$ exists as a real tensor if and only if all Fourier blocks are group invertible.
• Ring-Theoretic Result: The paper proves that the ring $(\mathbb{R}^{n \times n \times p}, +, *)$ is unit regular, meaning every element has an invertible von Neumann inverse.

Significance
The paper claims to provide a rigorous framework for transferring matrix-based constructions to real tensors while preserving the algebraic constraints of the t-product. Its primary significance lies in clarifying that real-valuedness in the t-product setting is a global property emerging from coordinated choices across conjugate Fourier blocks, rather than a local property of individual blocks. This distinction is essential for ensuring that theoretical constructions and computational algorithms for real tensors yield valid real results. The work bridges the gap between the complex algebraic theory of the t-product and the practical requirements of real-valued data applications.