Low T-Phase Rank Approximation of Third Order Tensors

We study low T-phase-rank approximation of sectorial third-order tensors $\mathscr{A}\in\mathbb{C}^{n\times n\times p}$ under the tensor T-product. We introduce canonical T-phases and T-phase rank, and formulate the approximation task as minimizing a symmetric gauge of the canonical phase vector under a T-phase-rank constraint. Our main tool is a tensor phase-majorization inequality for the geometric mean, obtained by lifting the matrix inequality through the block-circulant representation. In the positive-imaginary regime, this yields an exact optimal-value formula and an explicit optimal half-phase truncation family. We further establish tensor counterparts of classical matrix phase inequalities and derive a tensor small phase theorem for MIMO linear time-invariant systems.

💡 Research Summary

The paper develops a novel low‑T‑phase‑rank approximation theory for third‑order tensors under the T‑product, a framework that turns tensor multiplication into ordinary matrix multiplication via block‑circulant embedding and the discrete Fourier transform. After reviewing the T‑product algebra, the authors define sectorial tensors as those whose block‑circulant matrices lie entirely within a complex sector that does not contain the origin. For a sectorial tensor A, a sectorial decomposition A = Tᴴ * D * T exists, where D is a diagonal unitary tensor with entries e^{j ϕ_k(A)}. Ordering the phases ϕ_k(A) in non‑increasing order yields the canonical T‑phase vector ϕ(A); the number of non‑zero entries defines the T‑phase rank.

The central optimization problem is: given a prescribed T‑phase rank r, find a tensor E that minimizes a symmetric gauge Φ applied to its canonical phase vector, i.e., minimize Φ(ϕ(E)) subject to rank_T(E) ≤ r. The authors introduce the notion of a T‑phase gauge, which is any symmetric gauge function on ℝ^{np} (np being the total number of diagonal entries after block‑circulant lifting). This formulation parallels low‑rank approximation based on singular values but focuses on phase information.

The technical core is a phase‑majorization inequality for the tensor geometric mean. Building on Drury’s matrix inequality for accretive matrices, the authors prove that the block‑circulant embedding commutes with the T‑product geometric mean: bcirc(A # B) = bcirc(A) # bcirc(B). Consequently, the canonical phase vector of A # B is majorized by the component‑wise sum of the phase vectors of A and B. This result is lifted from the matrix setting to tensors via the block‑circulant representation, preserving the structure of the inequality.

In the “positive‑imaginary regime” (all phases lie in (0, π/2] or (−π/2, 0]), the authors construct an explicit optimal solution called the half‑phase truncation. The procedure keeps the largest r/2 phases unchanged and sets the remaining phases to zero, after globally sorting the phase vector. They prove that this truncation attains the exact minimum of Φ(ϕ(E)) for any symmetric gauge Φ, and they provide a closed‑form expression for the optimal objective value: Φ(ϕ_{r+1},…,ϕ_{np}). This mirrors the exact optimal‑value formula known for low‑rank approximation in the magnitude domain, but now applies to phase.

The paper then applies the phase calculus to tensor‑form MIMO linear time‑invariant (LTI) systems. When the system tensor is sectorial, a tensor small‑phase theorem is established: if all canonical phases lie within a sector of width less than π, the system is internally stable. This result offers a phase‑only stability criterion that complements traditional gain‑based methods such as H∞ or μ‑analysis, and it provides design guidance for controllers that must respect phase constraints.

Finally, the authors compare the phase‑based approximation with the classical T‑SVD low‑rank approximation. While T‑SVD minimizes error in a unitarily invariant norm by retaining the largest T‑singular values, the T‑phase approach minimizes a phase‑based gauge and can yield different approximations even with the same rank budget. They argue that in applications where phase information is critical—e.g., power‑grid dynamics, antenna array processing, and phase‑sensitive signal reconstruction—the T‑phase framework offers a more appropriate notion of low‑complexity representation.

Overall, the paper introduces a mathematically rigorous phase‑centric perspective on tensor approximation, provides exact optimality results in a significant regime, and demonstrates practical relevance through stability analysis of tensor‑based MIMO systems. Future work suggested includes extensions to nonlinear tensor dynamics, higher‑order tensors, and integration with randomized algorithms for large‑scale data.