Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization

TL;DR

The paper tackles approximate diagonalization and SVD-like Tucker decompositions of third-order tensors by introducing a Jacobi-type algorithm that operates on $2\times2\times2$ subtensors. It maximizes $\|\mathrm{diag}(\mathcal{A}\times_1U^T\times_2V^T\times_3W^T)\|^2$ over orthogonal $U,V,W$, using cyclic pivot updates with plane rotations whose angles are determined from local 2×2×2 subproblems via $\tan(2\phi)$ formulas. The authors prove convergence: every accumulation point of the iterates is a stationary point of the objective on $O_n\times O_n\times O_n$, and they provide the gradient structure to support the analysis. Numerical tests demonstrate convergence behavior across pivot strategies and initializations, discuss the impact of initialization (identity vs HOSVD) and parameter $\eta$, and address symmetric and antisymmetric tensor cases with preconditioning where needed. The method extends to higher-order tensors and offers a practical approach for approximate diagonalization and low-rank tensor approximation in applications such as signal processing and blind source separation.

Abstract

For a general third-order tensor $\mathcal{A}\in\mathbb{R}^{n\times n\times n}$ the paper studies two closely related problems, an SVD-like tensor decomposition and an (approximate) tensor diagonalization. We develop a Jacobi-type algorithm that works on $2\times2\times2$ subtensors and, in each iteration, maximizes the sum of squares of its diagonal entries. We show how the rotation angles are calculated and prove convergence of the algorithm. Different initializations of the algorithm are discussed, as well as the special cases of symmetric and antisymmetric tensors. The algorithm can be generalized to work on higher-order tensors.

Figures (4)

• Figure 1: Change in the relative off-norm for two $30\times30\times30$ tensors with different values of $\eta$. Left: Diagonalizable tensor. Right: Non-diagonalizable tensor.
• Figure 2: Change in the relative off-norm for six $10\times10\times10$ tensors with different pivot strategies.
• Figure 3: Convergence of the algorithm with different initializations. Left: $20\times20\times20$ tensor. Right: $10\times10\times10$ tensor.
• Figure 4: Departure from the symmetry for a random symmetric $20\times20\times20$ tensor.

Theorems & Definitions (3)

• proof
• proof
• proof : Proof of Theorem \ref{['tm:cvg']}